VARIATIONS ON A THEME OF JOST AND PAIS FRITZ GESZTESY, MARIUS MITREA, AND MAXIM ZINCHENKO

Abstract. We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schr¨ odinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schr¨ odinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set Ω ⊂ Rn , n ∈ N, n ≥ 2, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in L2 (Ω; dn x), n ∈ N, to modified Fredholm determinants associated with operators in L2 (∂Ω; dn−1 σ), n ≥ 2. Applications involving the Birman–Schwinger principle and eigenvalue counting functions are discussed.

1. Introduction

s1

To illustrate the reason behind the title of this paper, we briefly recall a celebrated result of Jost JP51 and Pais [43], who proved in 1951 a spectacular reduction of the Fredholm determinant associated with the Birman–Schwinger kernel of a one-dimensional Schr¨odinger operator on a half-line, to a simple Wronski determinant of distributional solutions of the underlying Schr¨odinger equation. This Wronski determinant also equals the so-called Jost function of the corresponding half-line Schr¨odinger operator. In this paper we prove aJP51 certain multi-dimensional variant of this result. D To describe the result due to Jost and Pais [43], we need a few preparations. Denoting by H0,+ N 2 and H0,+ the one-dimensional Dirichlet and Neumann Laplacians in L ((0, ∞); dx), and assuming V ∈ L1 ((0, ∞); dx), we introduce the perturbed Schr¨ odinger operators

D H+

and

(1.1) N H+

2

in L ((0, ∞); dx) by

D H+ f = −f 00 + V f,  D f ∈ dom H+ = {g ∈ L2 ((0, ∞); dx) | g, g 0 ∈ AC([0, R]) for all R > 0, 00

(1.2)

2

g(0) = 0, (−g + V g) ∈ L ((0, ∞); dx)}, N H+ f

00

= −f + V f,  N f ∈ dom H+ = {g ∈ L2 ((0, ∞); dx) | g, g 0 ∈ AC([0, R]) for all R > 0, 0

00

(1.3)

2

g (0) = 0, (−g + V g) ∈ L ((0, ∞); dx)}. Date: February 27, 2007. 2000 Mathematics Subject Classification. Primary: 47B10, 47G10, Secondary: 34B27, 34L40. Key words and phrases. Fredholm determinants, non-self-adjoint operators, multi-dimensional Schr¨ odinger operators, Dirichlet-to-Neumann maps. Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0405526, DMS-0400639, and FRG-0456306. To appear in J. Funct. Anal. 1

1.1

2

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

D N Thus, H+ and H+ are self-adjoint if and only if V is real-valued, but since the latter restriction plays no special role in our results, we will not assume real-valuedness of V throughout this paper s4 (except at the end of Section 4). D A fundamental system of solutions φD + (z, ·), θ+ (z, ·), and the Jost solution f+ (z, ·) of

−ψ 00 (z, x) + V ψ(z, x) = zψ(z, x),

z ∈ C\{0}, x ≥ 0,

are then introduced via the standard Volterra integral equations Z x 0 −1/2 1/2 dx0 z −1/2 sin(z 1/2 (x − x0 ))V (x0 )φD φD (z, x) = z sin(z x) + + (z, x ), + 0 Z x D 1/2 0 −1/2 D θ+ (z, x) = cos(z x) + dx z sin(z 1/2 (x − x0 ))V (x0 )θ+ (z, x0 ), Z ∞ 0 iz 1/2 x f+ (z, x) = e − dx0 z −1/2 sin(z 1/2 (x − x0 ))V (x0 )f+ (z, x0 ),

(1.4)

1.4

(1.5) (1.6) (1.7)

1.7

x

z ∈ C\{0}, Im(z 1/2 ) ≥ 0, x ≥ 0. In addition, we introduce u = exp(i arg(V ))|V |1/2 ,

v = |V |1/2 , so that V = u v,

(1.8)

2

and denote by I+ the identity operator in L ((0, ∞); dx). Moreover, we denote by W (f, g)(x) = f (x)g 0 (x) − f 0 (x)g(x),

x ≥ 0,

(1.9)

1

the Wronskian of f and g, where f, g ∈ C ([0, ∞)). We also use the standard convention to abbreviate (with a slight abuse of notation) the operator of multiplication in L2 ((0, ∞); dx) by an element f ∈ L1loc ((0, ∞); dx) (and similarly in the higher-dimensional context later) by the same symbol f (rather than Mf , etc.). For additional notational conventions we refer to the paragraph at the end of this introduction. Then, the following results hold: t1.1

Theorem 1.1. Assume V ∈ L1 ((0, ∞); dx) and let z ∈ C\[0, ∞) with Im(z 1/2 ) > 0. Then, −1 −1 D − zI N − zI u H0,+ v, u H0,+ v ∈ B1 (L2 ((0, ∞); dx)) + +

(1.10)

and ∞

Z  −1  −1/2 D − zI v = 1 + z det I+ + u H0,+ +

dx sin(z 1/2 x)V (x)f+ (z, x)

0

= W (f+ (z, ·), φD + (z, ·)) = f+ (z, 0), Z ∞    −1 N − zI det I+ + u H0,+ v = 1 + iz −1/2 dx cos(z 1/2 x)V (x)f+ (z, x) +

(1.11)

1.11

0 D 0 W (f+ (z, ·), θ+ (z, ·)) f+ (z, 0) =− = . (1.12) iz 1/2 iz 1/2 1.11 JP51 Equation (1.11) is the modern formulation of the classical result due to JostGM03 and Pais [43] (cf. GM03 the detailed discussion in [32]). Performing calculations similar to Section 4 in [32] for the pair of 1.12 N N and H+ , one obtains the analogous result (1.12).DD78 operators H0,+ For similar considerations inLS77 the context of finite interval1.11 problems, 1.12 we refer to Dreyfus and Dym [23] and Levit and Smilansky [50]. We emphasize that (1.11) and (1.12) exhibit the remarkable fact that the Fredholm determinant 2 associated with trace class operators in the infinite-dimensional space L ((0, ∞); dx) is reduced to 1.4 a simple Wronski determinant of C-valued distributional solutions of ( 1.4). This fact goes back to JP51 GM03 Ne72 Ne80 Ne02 Si00 Si05 Jost and Pais [43] (see also [32], [61], [62], [63, Sect. 12.1.2], [81], [82, Proposition 5.7], and the extensive literature cited in these references). The principal aim of this paper is to explore the

1.12

VARIATIONS ON A THEME OF JOST AND PAIS

3

extent to which this fact may generalize to higher dimensions n ∈ N, n ≥ 2. While a straightforward 1.11 1.12 generalization of (1.11), (1.12) appears to be difficult, we will next derive a formula for the ratio of such determinants which indeed permits a direct extension to higher dimensions. For this purpose we introduce the boundary trace operators γD (Dirichlet trace) and γN (Neumann trace) which, in the current one-dimensional half-line situation, are just the functionals, ( ( C 1 ([0, ∞)) → C, C([0, ∞)) → C, (1.13) γN : γD : h 7→ −h0 (0). g 7→ g(0), D N N In addition, we denote by mD 0,+ , m+ , m0,+ , and m+ the Weyl–Titchmarsh m-functions corresponding D D N N to H0,+ , H+ , H0,+ , and H+ , respectively, that is, 1/2 mD , 0,+ (z) = iz

mD + (z) =

0 (z, 0) f+ , f+ (z, 0)

mN 0,+ (z) = − mN + (z) = −

1 = iz −1/2 , mD (z) 0,+ 1 mD + (z)

=−

f+ (z, 0) 0 (z, 0) . f+

(1.14)

1.14

(1.15)

1.15

We briefly recall the spectral theoretic significance of mD + in the special case where V is real-valued: It is a Herglotz function (i.e., it maps the open complex upper half-plane C+ analytically into itself) and the measure dρD + in its Herglotz representation is then the spectral measure of the operator D D . Similarly, mD and hence encodes all spectral information of H+ H+ + also encodes all spectral N D N information of H+ since −1/m+ = m+ is also a Herglotz function and the measure dρN + in its N . In particular, dρD Herglotz representation represents the spectral measure of the operator H+ + determine V a.e. on (0, ∞) by the inverse spectral approach of Gelfand (respectively,GL55 dρN + ) uniquely Si99 GS00 Re03 Ge07 and Levitan [28] or Simon [80], [33] (see also Remling [75] and Section 6 in the survey [29]). 1.11 Then we obtain the following result for the ratio of the perturbation determinants in (1.11) and 1.12 (1.12): t1.2

D ) with Im(z 1/2 ) > 0. Then, Theorem 1.2. Assume V ∈ L1 ((0, ∞); dx) and let z ∈ C\σ(H+  −1  N − zI det I+ + u H0,+ v +    −1 D − zI det I+ + u H0,+ v +     D − zI )−1 V γ (H N − zI )−1 ∗ = 1 − γN (H+ (1.16) + D + 0,+

=

W (f+ (z), φN + (z)) 1/2 iz W (f+ (z), φD + (z))

=

0 f+ (z, 0) 1/2 iz f+ (z, 0)

=

mN mD 0,+ (z) + (z) = . D m0,+ (z) mN + (z) 1.16

(1.17)

At first sight it may seem unusual to even attempt to derive (1.16) in the one-dimensional context 1.17 since (1.17) already yields the reduction of a Fredholm determinant to a simple Wronski determinant. s4 t4.1 1.16 However, we will see in Section 4 (cf. Theorem 4.2) that it is precisely (1.16) that permits a natural extension to dimensions n ∈ N, n ≥ 2. Moreover, the latter is also t4.2 instrumental in proving the 1.17 analog of (1.17) in terms of Dirichlet-to-Neumann maps (cf. Theorem 4.3). The proper multi-dimensional generalizations to Schr¨odinger operators in L2 (Ω; dn x), corresponding to an open set Ω ⊂ Rn with compact, nonempty boundary ∂Ω, more precisely, the proper operator-valued generalization of the Weyl–Titchmarsh function mD + (z) is then given by the D Dirichlet-to-Neumann map, denoted by M (z). This operator-valued map indeed plays a fundamenΩ 1.17 1.17) to the higher-dimensional case. In particular, under Hypothesis tal role in our extension of ( h2.6 p 2.6 on Ω and V (which regulates smoothness properties of ∂Ω and L -properties of V ), we will prove 1.16 1.17 s4 the following multi-dimensional extension of (1.16) and (1.17) in Section 4:

1.16 1.17

4

t1.3

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

   h2.6 D Theorem 1.3. Assume Hypothesis 2.6 and let k ∈ N, k ≥ p and z ∈ C σ HΩD ∪ σ H0,Ω ∪  N σ H0,Ω . Then,  −1  N − zI det k IΩ + u H0,Ω v Ω  −1  D − zI det k IΩ + u H0,Ω v Ω    −1  N − zI )−1 ∗ etr(Tk (z)) = det k I∂Ω − γN HΩD − zIΩ (1.18) V γD (H0,Ω Ω  D = det k MΩD (z)M0,Ω (z)−1 etr(Tk (z)) . (1.19) Here, detk (·) denotes the modified Fredholm determinant in connection with Bk perturbations of the identity and Tk (z) is some trace class operator. In particular, T2 (z) is given by −1  −1 −1 ∗ N − zI D − zI V γD H0,Ω T2 (z) = γN H0,Ω V HΩD − zIΩ , (1.20) Ω Ω where IΩ and I∂Ω represent the identity operators in L2 (Ω; dn x) and L2 (∂Ω; dn−1 σ), respectively (with dn−1 σ denoting the surface measure on ∂Ω). The sudden appearance of the term exp(tr(Tk (z))) 1.18 1.19 in (1.18) and (1.19), when compared to thet1.3 one-dimensional case, is due to the necessary use of the modified determinant detk (·) in Theorem 1.3. 1.18 1.16 We note that the multi-dimensional extension (1.18) of (1.16), under the stronger hypothesis GLMZ05 t1.3 V ∈ L2 (Ω; dn x), n = 2, 3, first appeared in [31]. However, the present results in Theorem 1.3 GLMZ05 go decidedlyh2.6 beyond those in [31] in the following sense: (i) the class of domains Ω permitted by h2.1 GLMZ05 Hypothesis 2.6 (actually, Hypothesis 2.1) is greatly enlarged as compared to [31]; (ii) the multi1.19 1.17 Dirichlet-to-Neumann maps is a new (and the most dimensional extension (1.19) of (1.17) invoking GLMZ05 significant) result in this paper; (iii) while [31] focused on dimensions n = 2, 3, we now treat the general case n ∈ N, s4 n ≥ 2; (iv) we provide an application involving eigenvalue counting functions at the end of Section 4; (v) we study a representation of the product formula for modified Fredholm s4 4. determinants, which should be of independent interest, at the beginning of Section t1.3 The principal reduction in Theorem 1.3 reduces (a ratio of) 1.18 modified Fredholm determinants associated with operators in L2 (Ω; dn x) on the left-hand side of (1.18) to modified Fredholm deter1.18 2 n−1 minants associated with operators in L (∂Ω; d σ) on the right-hand side of ( 1.18) and especially, 1.19 in ( 1.19). This is the analog of the reduction described in the one-dimensional context of Theorem t1.2 1.2, where Ω corresponds to the half-line (0, ∞) and its boundary ∂Ω corresponds to the one-point 1.16 set {0}. As a result, the ratio of determinants on the left-hand side of (1.16) associated with operators in L2 ((0, ∞);1.16 dx) is reduced to ratios of Wronskians and Weyl–Titchmarsh functions on the 1.17 right-hand side of (1.16) and in (1.17). Finally, we briefly list most of the notational conventions used throughout this paper. Let H be a separable complex Hilbert space, (·, ·)H the scalar product in H (linear in the second factor), and IH the identity operator in H. Next, let T be a linear operator mapping (a subspace of) a Banach space into another, with dom(T ) and ran(T ) denoting the domain and range of T . The closure of a closable operator S is denoted by S. The kernel (null space) of T is denoted by ker(T ). The spectrum and resolvent set of a closed linear operator in H will be denoted by σ(·) and ρ(·). The Banach spaces of bounded and compact linear operators in H are denoted by B(H) and B∞ (H), respectively. Similarly, the Schatten–von Neumann (trace) ideals will subsequently be denoted by Bk (H), k ∈ N. Analogous notation B(H1 , H2 ), B∞ (H1 , H2 ), etc., will be used for bounded, compact, etc., operators between two Hilbert spaces H1 and H2 . In addition, tr(T ) denotes the trace of a trace class operator T ∈ B1 (H) and detp (IH + S) represents the (modified) Fredholm determinant associated with an operator S ∈ Bk (H), k ∈ N (for k = 1 we omit the subscript 1). Moreover, X1 ,→ X2 denotes the continuous embedding of the Banach space X1 into the Banach space X2 .

1.18 1.19

VARIATIONS ON A THEME OF JOST AND PAIS

5

For general references on the theory ofGK69 (modified) Fredholm determinants we refer, for instance, DS88 GGK00 RS78 Si77 Si05 to [24, Sect. XI.9], [34, Ch. Chs. IX–XI], [35, Ch. Sect. 4.2], [74, Sect. XIII.17], [79], and [82, Ch. 9].

s2

¨ dinger Operators with Dirichlet and Neumann boundary conditions 2. Schro D N In this section we primarily focus on various properties of Dirichlet, H0,Ω , and Neumann, H0,Ω , 2 n n Laplacians in L (Ω; d x) associated with open sets Ω ⊂ R , n ∈ N, n ≥ 2, introduced in Hypothesis −q h2.1 D,N 2.1 below. In particular, we study mapping properties of H0,Ω − zIΩ , q ∈ [0, 1] (with IΩ −q D,N 2 n the identity operator in L (Ω; d x)) and trace ideal properties of the maps f H0,Ω − zIΩ ,   −r −s D N f ∈ Lp (Ω; dn x), for appropriate p ≥ 2, and γN H0,Ω −zIΩ , and γD H0,Ω −zIΩ , for appropriate r > 2.2 3/4, s > 1/4, with γN and γD being the Neumann and Dirichlet boundary trace operators defined 2.3 in (2.2) and (2.3). At the end of this section we then introduce the Dirichlet and Neumann Schr¨odinger operators D HΩD and HΩN in L2 (Ω; dn x), that is, perturbations of the Dirichlet and Neumann Laplacians H0,Ω h2.6 N and H0,Ω by a potential V satisfying Hypothesis 2.6. We start with introducing our assumptions on the set Ω:

h2.1

Hypothesis 2.1. Let n ∈ N, n ≥ 2, and assume that Ω ⊂ Rn is an open set with a compact, nonempty boundary ∂Ω. In addition, we assume that one of the following three conditions holds: (i) Ω is of class C 1,r for some 1/2 < r < 1; (ii) Ω is convex; (iii) Ω is a Lipschitz domain satisfying a uniform exterior ball condition (UEBC ). We note that while ∂Ω is assumed to be compact, Ω may be unbounded in connection with sA conditions (i) or (iii). For more details in this context we refer to Appendix A. 0 (Dirichlet trace) by First, we introduce the boundary trace operator γD 0 γD : C(Ω) → C(∂Ω),

0 γD u = u|∂Ω .

(2.1)

Mc00

Then there exists a bounded, linear operator γD (cf. [53, Theorem 3.38]), γD : H s (Ω) → H s−(1/2) (∂Ω) ,→ L2 (∂Ω; dn−1 σ), γD : H

3/2

(Ω) → H

1−ε

2

(∂Ω) ,→ L (∂Ω; d

n−1

σ),

1/2 < s < 3/2,

(2.2)

ε ∈ (0, 1),

2.2

0 whose action is compatible with that of γD . That is, the two Dirichlet trace operators coincide on the intersection of sA their domains. We recall that dn−1 σ denotes the surface measure on ∂Ω and we refer to Appendix A for our notation in connection with Sobolev spaces. Next, we introduce the operator γN (Neumann trace) by

γN = ν · γD ∇ : H s+1 (Ω) → L2 (∂Ω; dn−1 σ),

1/2 < s < 3/2,

(2.3)

2.3

2.2

where ν denotes the outward pointing normal unit vector to ∂Ω. It follows from (2.2) that γN is also a bounded operator. h2.1 Given Hypothesis 2.1, we introduce the self-adjoint and nonnegative Dirichlet and Neumann D N Laplacians H0,Ω and H0,Ω associated with the domain Ω as follows,  D D H0,Ω = −∆, dom H0,Ω = {u ∈ H 2 (Ω) | γD u = 0}, (2.4)  N N H0,Ω = −∆, dom H0,Ω = {u ∈ H 2 (Ω) | γN u = 0}. (2.5) sA

D N A detailed discussion of H0,Ω and H0,Ω is provided in Appendix A.

2.4 2.5

6

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

h2.1

l2.2

2.4

D N Lemma 2.2. Assume Hypothesis 2.1. Then the operators H0,Ω and H0,Ω introduced in (2.4) and 2.5 2 n (2.5) are nonnegative and self-adjoint in L (Ω; d x) and the following boundedness properties hold for all q ∈ [0, 1] and z ∈ C\[0, ∞),  −q −q D N H0,Ω − zIΩ , H0,Ω − zIΩ ∈ B L2 (Ω; dn x), H 2q (Ω) . (2.6)

2.6

2.6

The fractional powers in (2.6) (and in subsequent analogous cases) are defined via the functional calculus implied by the spectral theorem for self-adjointlA.2 operators. sA A (cf. particularly Lemma A.2), the key ingredients in proving Lemma As explained in Appendix l2.2 2.2 are the inclusions   D N dom H0,Ω ⊂ H 2 (Ω), dom H0,Ω ⊂ H 2 (Ω) (2.7) and methods based on real interpolation spaces. For the remainder of this paper we agree to the simplified notation that the operator of multiplication by the measurable function f in GLMZ05 L2 (Ω; dn x) is again denoted by the symbol f . The next result is an extension of [31, Lemma 6.8] and aims at an explicit discussion of the GLMZ05 z-dependence of the constant c appearing in estimate (6.48) of [31]. h2.1

l2.3

Lemma 2.3. Assume Hypothesis 2.1 and let 2 ≤ p, n/(2p) < q ≤ 1, f ∈ Lp (Ω; dn x), and z ∈ C\[0, ∞). Then, −q −q  D N f H0,Ω − zIΩ , f H0,Ω − zIΩ ∈ Bp L2 (Ω; dn x) , (2.8) and for some c > 0 (independent of z and f )

−q 2 D

f H0,Ω − zIΩ Bp (L2 (Ω;dn x))   |z|2q + 1 2 −q 2 kLp (Rn ;dn x) kf k2Lp (Ω;dn x) , ≤c 1+  2q k(|·| − z) D dist z, σ H0,Ω

−q 2 N

f H0,Ω

− zIΩ Bp (L2 (Ω;dn x))   |z|2q + 1 2 −q 2 2 ≤c 1+ 2q k(|·| − z) kLp (Rn ;dn x) kf kLp (Ω;dn x) . N dist z, σ H0,Ω

2.7

(2.9)

2.8

Proof. We start by noting that under the assumption that Ω is a Lipschitz domain, there is a bounded extension operator E,  E ∈ B H s (Ω), H s (Rn ) such that (Eu)|Ω = u, u ∈ H s (Ω), (2.10)

2.9

Ry99

for all s ∈ R (see, e.g., [76]). Next, for notational convenience, we denote by H0,Ω either one of the D N operators H0,Ω or H0,Ω and by RΩ the restriction operator ( L2 (Rn ; dn x) → L2 (Ω; dn x), RΩ : (2.11) u 7→ u|Ω . Moreover, we introduce the following extension f˜ of f , ( f (x), x ∈ Ω, ˜ f (x) = f˜ ∈ Lp (Rn ; dn x). 0, x ∈ Rn \Ω,

(2.12)

Then, f (H0,Ω − zIΩ )−q = RΩ f˜(H0 − zI)−q (H0 − zI)q E(H0,Ω − zIΩ )−q , (2.13) 2 n n where (for simplicity) I denotes the identity operator in L (R ; d x) and H0 denotes the nonnegative self-adjoint operator H0 = −∆, dom(H0 ) = H 2 (Rn ) (2.14)

2.12

VARIATIONS ON A THEME OF JOST AND PAIS

7

in L2 (Rn ; dn x). lA.2 Let g ∈ L2 (Ω; dn x) and define h = (H0,Ω − zIΩ )−q g. Then by Lemma A.2, h ∈ H 2q (Ω) ⊂ L2 (Ω; dn x). Using the spectral theorem for the nonnegative self-adjoint operator H0,Ω in L2 (Ω; dn x), one computes,

2 2 khkL2 (Ω;dn x) = (H0,Ω − zIΩ )−q g L2 (Ω;dn x) Z  = (2.15) |λ − z|−2q dEH0,Ω (λ)g, g L2 (Ω;dn x)

2.14

σ(H0,Ω )

2

≤ dist(z, σ(H0,Ω ))−2q kgkL2 (Ω;dn x) and since (H0,Ω + IΩ )−q ∈ B(L2 (Ω; dn x), H 2q (Ω)),

2 2 2 khkH 2q (Ω) = (H0,Ω + IΩ )−q (H0,Ω + IΩ )q h H 2q (Ω) ≤ c k(H0,Ω + IΩ )q hkL2 (Ω;dn x) Z  =c |λ + 1|2q dEH0,Ω (λ)h, h L2 (Ω;dn x) σ(H0,Ω )

Z ≤ 2c σ(H0,Ω )

  |λ − z|2q + |z + 1|2q dEH0,Ω (λ)h, h L2 (Ω;dn x) 2

2

= 2c k(H0,Ω − zIΩ )q hkH 2q (Ω) + |z + 1|2q khkL2 (Ω;dn x)  2 ≤ 2c 1 + |z + 1|2q dist(z, σ(H0,Ω ))−2q kgkL2 (Ω;dn x) ,

(2.16)

2.15



where EH0,Ω (·) denotes the family of spectral projections of H0,Ω . Moreover, utilizing the represenq tation of (H0 − zI)q as2.9 the operator of multiplication by |ξ|2 − z in the Fourier space L2 (Rn ; dn ξ), and the fact that by (2.10)   E ∈ B H 2q (Ω), H 2q (Rn ) ∩ B L2 (Ω; dn x), L2 (Rn ; dn x) , (2.17) one computes q

k(H0 − zI)

2 EhkL2 (Rn ;dn x)

Z

2q 2 c dn ξ |ξ|2 − z |(Eh)(ξ)|

= Rn

Z ≤2

 2 c dn ξ |ξ|4q + |z|2q |(Eh)(ξ)|

Rn 2

2

≤ 2 kEhkH 2q (Rn ) + |z|2q kEhkL2 (Rn ;dn x)  2 2 ≤ 2c khkH 2q (Ω) + |z|2q khkL2 (Ω;dn x) . 2.14

2.15

(2.18)

2.17

(2.19)

2.18

(2.20)

2.19

(2.21)

2.20

(2.22)

2.21



2.17

Combining the estimates (2.15), (2.16), and (2.18), one obtains  (H0 − zI)q E(H0,Ω − zIΩ )−q ∈ B L2 (Ω; dn x), L2 (Rn ; dn x) and the following norm estimate with some constant c > 0,

(H0 − zI)q E(H0,Ω − zIΩ )−q 2 2 ≤c+ B(L (Ω;dn x),L2 (Rn ;dn x)) Si05

c(|z|2q + 1) . dist(z, σ(H0,Ω ))2q

RS79

Next, by [82, Theorem 4.1] (or [73, Theorem XI.20]) one obtains  f˜(H0 − zI)−q ∈ Bp L2 (Rn ; dn x) and

f˜(H0 − zI)−q B

2 n n p (L (R ;d x))

≤ c k(|·|2 − z)−q kLp (Rn ;dn x) kf˜kLp (Rn ;dn x) = c k(|·|2 − z)−q kLp (Rn ;dn x) kf kLp (Ω;dn x) .

8

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

2.7

2.12

2.18

2.20

2.8

2.12

2.19

2.21

Thus, (2.8) follows from (2.13), (2.19), (2.21), and (2.9) follows from (2.13), (2.20), and (2.22).



Next we recall certain mapping properties of powers of the resolvents of Dirichlet and Neumann Laplacians multiplied by the Neumann and Dirichlet boundary trace operators, respectively: h2.1

l2.4

Lemma 2.4. Assume Hypothesis 2.1 and let ε > 0, z ∈ C\[0, ∞). Then, −(3+ε)/4  −(1+ε)/4 D N γN H0,Ω − zIΩ , γD H0,Ω − zIΩ ∈ B L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) . GLMZ05

l2.4

l2.2

2.2

(2.23)

2.22

2.3

As in [31, Lemma 6.9], Lemma 2.4 follows from Lemma 2.2 and from (2.2) and (2.3). h2.1

c2.5

Corollary 2.5. Assume Hypothesis 2.1 and let f1 ∈ Lp1 (Ω; dn x), p1 ≥ 2, p1 > 2n/3, f2 ∈ Lp2 (Ω; dn x), p2 > 2n, and z ∈ C\[0, ∞). Then, denoting by f1 and f2 the operators of multiplication by functions f1 and f2 in L2 (Ω; dn x), respectively, one has −1  N − zI γD H0,Ω f1 ∈ Bp1 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , (2.24) Ω   −1 D − zI γN H0,Ω f2 ∈ Bp2 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) (2.25) Ω and for some cj (z) > 0 (independent of fj ), j = 1, 2,

−1

N − zI f1 ≤ c1 (z) kf1 kLp1 (Ω;dn x) ,

γD H0,Ω Ω Bp1 (L2 (Ω;dn x),L2 (∂Ω;dn−1 σ))

−1

D − zI f2 ≤ c2 (z) kf2 kLp2 (Ω;dn x) .

γN H0,Ω Ω 2 n 2 n−1 Bp2 (L (Ω;d x),L (∂Ω;d

GLMZ05

c2.5

2.25 2.26

(2.26)

2.27

(2.27)

2.28

σ))

l2.3

l2.4

As in [31, Corollary 6.10], Corollary 2.5 follows from Lemmas 2.3 and 2.4. Finally, we turn to our assumptions on the potential V and the corresponding definition of Dirichlet and Neumann Schr¨ odinger operators HΩD and HΩN in L2 (Ω; dn x): h2.1

h2.6

Hypothesis 2.6. Suppose that Ω satisfies Hypothesis 2.1 and assume that V ∈ Lp (Ω; dn x) for some p satisfying p > 4/3 in the case n = 2, and p > n/2 in the case n ≥ 3. h2.6

Assuming Hypothesis 2.6, we next introduce the perturbed operators H D and HΩN in L2 (Ω; dn x) sB Ω by alluding to abstract perturbation results summarized in Appendix B as follows: Let V , u, and v denote the operators of multiplication by functions V , u =l2.3 exp(i arg(V ))|V |1/2 , and v = |V |1/2 in 2 n 2p n L (Ω; d x), respectively. Since u, v ∈ L (Ω; d x), Lemma 2.3 yields −1/2 −1/2  D D − zI u H0,Ω − zIΩ , H0,Ω v ∈ B2p L2 (Ω; dn x) , z ∈ C\[0, ∞), (2.28) Ω    −1/2 −1/2 N N − zI u H0,Ω − zIΩ , H0,Ω v ∈ B2p L2 (Ω; dn x) , z ∈ C\[0, ∞), (2.29) Ω

2.31 2.32

and hence, in particular,  N dom(u) = dom(v) ⊇ H 1 (Ω) ⊃ H 2 (Ω) ⊃ dom H0,Ω ,  D dom(u) = dom(v) ⊇ H 1 (Ω) ⊇ H01 (Ω) ⊃ dom H0,Ω . D H0,Ω ,

N H0,Ω ,

Thus, the operators 2.31 2.32 (2.28) and (2.29) imply

D − zI u H0,Ω Ω

−1

(2.31) sB

u, and v satisfy Hypothesis B.1 (i) (see Appendix B). Moreover,

N − zI v, u H0,Ω Ω

hB.1

hB.1

(2.30)

−1

 v ∈ Bp L2 (Ω; dn x) , 2.8

z ∈ C\[0, ∞), l2.3

(2.32)

D N which verifies Hypothesis B.1 (ii) for H0,Ω and H0,Ω . Utilizing (2.9) in Lemma 2.3 with −z > 0 2.31 2.32 sufficiently large, such that the B2p -norms of the operators in (2.28) and (2.29) are less than 1, 2.35 and hence the Bp -norms of the operators in (2.32) are less than 1, one also verifies Hypothesis hB.1 tB.2 D B.1 (iii). Thus, applying Theorem B.2 one obtains closed operators  the densely defined,  HΩ and N D D N N HΩ (which are extensions of H0,Ω +V on dom H0,Ω ∩dom(V ) and H0,Ω +V on dom H0,Ω ∩dom(V ),

2.35

VARIATIONS ON A THEME OF JOST AND PAIS

9

respectively). In particular, the resolvent of HΩD (respectively, HΩN ) is explicitly given by the analog B.5 D N of (B.5) in terms of the resolvent of H0,Ω (respectively, H0,Ω ) and the factorization V = uv. 2.6 2.8 2.22 2.25 2.28 2.31 2.32 2.35 We note in passing that (2.6)–(2.9), (2.23), (2.24)–(2.27), (2.28), (2.29), (2.32), etc., extend of D N course to all z in the resolvent set of the corresponding operators H0,Ω and H0,Ω .

s3

3. Dirichlet and Neumann boundary value problems and Dirichlet-to-Neumann maps This section is devoted to Dirichlet and Neumann boundary value problems associated with the Helmholtz differential expression −∆ − z as well as the corresponding differential expression −∆ + V − z in the presence of a potential V , both in connection with the open set Ω. In addition, D we provide a detailed discussion of Dirichlet-to-Neumann, M0,Ω , MΩD , and Neumann-to-Dirichlet N N 2 n−1 maps, M0,Ω , MΩ , in L (∂Ω; d σ). Denote by ∗  → H −1/2 (∂Ω) (3.1) γ eN : u ∈ H 1 (Ω) ∆u ∈ H 1 (Ω) a weak Neumann trace operator defined by Z he γN u, φi = dn x ∇u(x) · ∇Φ(x) + h∆u, Φi (3.2)

3.0

3.1a



t3.1

for all φ ∈ H 1/2 (∂Ω) and Φ ∈ H 1 (Ω) such that γD Φ = φ. We note that this definition is independent of the particular2.3 extension Φ of φ, and that γ eN is a bounded extension of the Neumann trace operator A.11 A.16 γN defined in (2.3). For more details we refer to equations (A.14)–(A.17). We start with the Helmholtz Dirichlet and Neumann boundary value problems:   h2.1 D Theorem 3.1. Assume Hypothesis 2.1. Then for every f ∈ H 1 (∂Ω) and z ∈ C σ H0,Ω the following Dirichlet boundary value problem, ( 3/2 (−∆ − z)uD uD (Ω), 0 = 0 on Ω, 0 ∈H (3.3) D γD u0 = f on ∂Ω, 2 n−1 has a unique solution uD eN uD σ). Moreover, there exist constants C D = 0 satisfying γ 0 ∈ L (∂Ω; d D C (Ω, z) > 0 such that D kuD (3.4) 0 kH 3/2 (Ω) ≤ C kf kH 1 (∂Ω) .  2 n−1 N Similarly, for every g ∈ L (∂Ω; d σ) and z ∈ C\σ H0,Ω the following Neumann boundary value problem, ( 3/2 (−∆ − z)uN uN (Ω), 0 = 0 on Ω, 0 ∈H (3.5) N γ eN u0 = g on ∂Ω, N 1 N = has a unique solution uN 0 satisfying γD u0 ∈ H (∂Ω). Moreover, there exist constants C N C (Ω, z) > 0 such that N kuN (3.6) 0 kH 3/2 (Ω) ≤ C kgkL2 (∂Ω;dn−1 σ) . 3.1 3.4a In addition, (3.3)–(3.6) imply that the following maps are bounded  −1 ∗ ∗   D D γN H0,Ω − zIΩ : H 1 (∂Ω) → H 3/2 (Ω), z ∈ C σ H0,Ω , (3.7)  −1 ∗ ∗   N 2 n−1 3/2 N γD H0,Ω − zIΩ : L (∂Ω; d σ) → H (Ω), z ∈ C σ H0,Ω . (3.8)

3.1

3.3a

3.2

3.4a

3.4b 3.4c

N Finally, the solutions uD 0 and u0 are given by the formulas

−1 ∗ D uD f, 0 (z) = − γN H0,Ω − zIΩ   −1 ∗ N uN g. 0 (z) = γD H0,Ω − zIΩ

(3.9)

3.3

(3.10)

3.4

10

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

Mi96

Proof. It follows from Theorem 9.3 in [56] that the boundary value problems, ( 2 n−1 (∆ + z)uD N (∇uD σ), 0 = 0 on Ω, 0 ) ∈ L (∂Ω; d D 1 γD u0 = f ∈ H (∂Ω) on ∂Ω

(3.11)

3.5

and ( 2 n−1 (∆ + z)uN N (∇uN σ), 0 = 0 on Ω, 0 ) ∈ L (∂Ω; d (3.12) N 2 n−1 γ eN u0 = g ∈ L (∂Ω; d σ) on ∂Ω,   D N have unique solutions for all z ∈ C\σ H0,Ω and z ∈ C\σ H0,Ω , respectively, satisfying natural JK95 Mi96 estimates. Here N (·) denotes the non-tangential maximal function (cf. [42], [56]) (N w)(x) = sup |w(y)|,

x ∈ ∂Ω,

3.6

(3.13)

y∈Γ(x)

where w is a locally bounded function and Γ(x) is a nontangential approach region with vertex at x, that is, for some fixed constant C > 1 one has Γ(x) = {y ∈ Ω | |x − y| < C dist(y, ∂Ω)}.

(3.14) JK95

In the case of a bounded domain Ω, it follows from Corollary 5.7 in [42] that for any harmonic function v in Ω, N (∇v) ∈ L2 (∂Ω; dn−1 σ) if and only if v ∈ H 3/2 (Ω),

(3.15)

3.7

accompanied with natural estimates. For any solution u of the Helmholtz equation (∆ + z)u = 0 on a bounded domain Ω, one can introduce the harmonic function Z v(x) = u(x) + z dn y En (x − y)u(y), x ∈ Ω, (3.16) Ω 2

n−1

such that N (∇u) ∈ L (∂Ω; d σ) if and only if N (∇v) ∈ L2 (∂Ω; dn−1 σ), and u ∈ H 3/2 (Ω) if 3/2 and only if v ∈ H (Ω). (Again, natural estimates are valid in each case.) Here En denotes the fundamental solution of the Laplace equation in Rn , n ∈ N, n ≥ 2, ( 1 ln(|x|), n = 2, , x ∈ Rn \{0}, (3.17) En (x) = 2π 1 2−n |x| , n ≥ 3, (2−n)ωn−1 3.7

with ωn−1 denoting the area of the unit sphere in Rn . The equivalence in (3.15) extends from harmonic functions to all functions u satisfying the Helmholtz equation, (∆ + z)u = 0 on a bounded domain Ω, N (∇u) ∈ L2 (∂Ω; dn−1 σ) if and only if u ∈ H 3/2 (Ω). 3.1

3.2

3.5

(3.18) 3.6

3.8

Thus, in the case of a bounded domain Ω, (3.3) and (3.5) follow from (3.11), (3.12), and (3.18). Moreover, one has the chain of estimates    D

kuD + kuD (3.19) 0 kH 3/2 (Ω) ≤ C1 N ∇u0 0 kL2 (Ω;dn x) ≤ C2 kf kH 1 (L2 (∂Ω;dn−1 σ)) L2 (∂Ω;dn−1 σ) for some constants Ck > 0, k = 1, 2. In the case of an unbounded domain Ω, one first obtains   3.8 D (3.18) for Ω ∩ B, where B is a sufficiently large ball containing ∂Ω. Then, since z ∈ C σ H0,Ω =   N C σ H0,Ω = C\[0, ∞) (since now Ω contains the exterior of a ball in Rn ), one exploits the exponen3.8 tial decay of solutions of the Helmholtz equation to extend (3.18) from Ω ∩ B to Ω. This, together 3.5 3.6 3.1 3.2 with (3.11) and (3.12), yields (3.3)3.3 and (3.5). 3.4 l2.4 Next, we turn to the proof of (3.9) and (3.10). We note that by Lemma 2.4, −1 −1  N D γ eN H0,Ω − zIΩ , γD H0,Ω − zIΩ ∈ B L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , (3.20)

3.8

VARIATIONS ON A THEME OF JOST AND PAIS

11

and hence D γ eN H0,Ω − zIΩ

−1 ∗

N , γD H0,Ω − zIΩ

−1 ∗

 ∈ B L2 (∂Ω; dn−1 σ), L2 (Ω; dn x) . 3.1

(3.21)

3.21a

3.2

N Then, denoting by uD 0 and u0 the unique solutions of (3.3) and (3.5), respectively, and using Green’s formula, one computes −1   D D v L2 (Ω;dn x) uD 0 , v L2 (Ω;dn x) = u0 , (−∆ − z) H0,Ω − zIΩ  −1  D = (−∆ − z)uD v L2 (Ω;dn x) 0 , H0,Ω − zIΩ   −1 D v L2 (∂Ω;dn−1 σ) + γ eN uD 0 , γD H0,Ω − zIΩ −1

D − zI − γ v, γD uD eN H0,Ω Ω 0 , 1/2

−1 D − zI =− γ v, f 1/2 eN H0,Ω Ω −1 ∗  D =− γ eN H0,Ω − zIΩ f, v L2 (Ω;dn x) (3.22)

and uN 0 ,v

 L2 (Ω;dn x)

−1  v L2 (Ω;dn x)   −1 N = (−∆ − z)uN v L2 (Ω;dn x) 0 , H0,Ω − zIΩ −1  N v L2 (∂Ω;dn−1 σ) + γ eN uN 0 , γD H0,Ω − zIΩ

 −1 N − zI − γ eN H0,Ω v, γD uN Ω 0 1/2 −1  N v L2 (∂Ω;dn−1 σ) = g, γD H0,Ω − zIΩ    −1 ∗ N g, v L2 (Ω;dn x) = γD H0,Ω − zIΩ N = uN 0 , (−∆ − z) H0,Ω − zIΩ

3.3 This proves 3.4a (3.9) 3.3a

(3.23)

3.4

for any v 3.21a ∈ L2 (Ω; dn x). and (3.10) with the3.4b operators3.4c involved understood in the sense of (3.21). Granted (3.4) and (3.6), one finally obtains (3.7) and (3.8).  We temporarily strengthen our hypothesis on V and introduce the following assumption: h2.1

h3.2

Hypothesis 3.2. Suppose the set Ω satisfies Hypothesis 2.1 and assume that V ∈ Lp (Ω; dn x) for some p > 2 if n = 2, 3 and p ≥ 2n/3 if n ≥ 4.

t3.3

By employing a perturbative approach, we now extend Theorem 3.1 in connection with the Helmholtz differential expression −∆ − z on Ω to the case of a Schr¨odinger differential expression −∆ + V − z on Ω.   h3.2 Theorem 3.3. Assume Hypothesis 3.2. Then for every f ∈ H 1 (∂Ω) and z ∈ C σ HΩD the following Dirichlet boundary value problem, ( (−∆ + V − z)uD = 0 on Ω, uD ∈ H 3/2 (Ω), (3.24) γD uD = f on ∂Ω,

t3.1

3.9

has a unique solution uD satisfying γ eN uD ∈ L2 (∂Ω; dn−1 σ). Moreover, there exist constants C D = D C (Ω, z) > 0 such that kuD kH 3/2 (Ω) ≤ C D kf kH 1 (∂Ω) .

(3.25)

3.9a

12

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

  Similarly, for every g ∈ L2 (∂Ω; dn−1 σ) and z ∈ C σ HΩN the following Neumann boundary value problem, ( (−∆ + V − z)uN = 0 on Ω, uN ∈ H 3/2 (Ω), (3.26) γ eN uN = g on ∂Ω,

3.10

has a unique solution uN . Moreover, there exist constants C N = C N (Ω, z) > 0 such that kuN kH 3/2 (Ω) ≤ C N kgkL2 (∂Ω;dn−1 σ) . 3.9

(3.27)

3.10a

(3.28)

3.10b

(3.29)

3.10c

3.10a

In addition, (3.24)–(3.27) imply that the following maps are bounded    −1 ∗ ∗ γN HΩD − zIΩ : H 1 (∂Ω) → H 3/2 (Ω), z ∈ C σ HΩD ,  −1 ∗ ∗   γD HΩN − zIΩ : L2 (∂Ω; dn−1 σ) → H 3/2 (Ω), z ∈ C σ HΩN . Finally, the solutions u

D

and u

N

are given by the formulas  −1 ∗ ∗ u (z) = − γN HΩD − zIΩ f, (3.30)  −1 ∗ ∗ N N u (z) = γD HΩ − zIΩ g. (3.31)    D ∪ σ HΩD in the case of the Dirichlet problem Proof. We temporarily assume that z ∈ C σ H0,Ω    N and z ∈ C σ H0,Ω ∪ σ HΩN in the context of the Neumann problem. Uniqueness of solutions / σ(HΩN ), respectively. follows from the fact that z ∈ / σ(HΩD ) and z ∈ Next, we will show that the functions −1 D D uD (z) = uD V u0 (z), (3.32) 0 (z) − HΩ − zIΩ  −1 N uN (z) = uN V uN (3.33) 0 (z) − HΩ − zIΩ 0 (z), D

t3.1

3.11

3.11 3.12

3.13 3.14

3.12

N with uD 0 , ut3.1 0 given by Theorem 3.1, satisfy (3.30) and (3.31), respectively. Indeed, it follows from N 3/2 2 n−1 (Ω) and γ eN uD σ). Using the Sobolev embedding Theorem 3.1 that uD 0 , u0 ∈ H 0 ∈ L (∂Ω; d theorem

H 3/2 (Ω) ,→ Lq (Ω; dn x) for all q ≥ 2 if n = 2, 3 and 2 ≤ q ≤ 2n/(n − 3) if n ≥ 4, and the fact that V ∈ Lp (Ω; dn x), p > 2 if n = 3.14 2, 3 and 3.13 N 2 n V uD , V u ∈ L (Ω; d x), and hence ( 3.32) and ( 3.33) are 0 0 −1 −1 l2.3 N D − zIΩ ∈ Bp − zIΩ , V H0,Ω Lemma 2.3 that V H0,Ω  −1 −1  D IΩ + V H0,Ω − zIΩ ∈ B L2 (Ω; dn x) ,  −1 −1  N IΩ + V H0,Ω − zIΩ ∈ B L2 (Ω; dn x) , tB.3

2.4

p ≥ 2n/3 if n ≥ 4, one concludes that well-defined. Moreover, it follows from  L2 (Ω; dn x) , and hence    D z ∈ C σ H0,Ω ∪ σ HΩD , (3.34)    N z ∈ C σ H0,Ω ∪ σ HΩN , (3.35)

2.5

by applying Theorem B.3. Thus, by (2.4) and (2.5), −1 D −1  −1 −1 D D D HΩD − zIΩ V u0 = H0,Ω − zIΩ IΩ + V H0,Ω − zIΩ V u0 ∈ H 2 (Ω), −1 N −1  −1 −1 N N N HΩN − zIΩ V u0 = H0,Ω − zIΩ IΩ + V H0,Ω − zIΩ V u0 ∈ H 2 (Ω), D

and hence u , u

N

∈H

3/2

(Ω) and γ eN u

D

2

∈ L (∂Ω; d

n−1

(3.37)

σ). Moreover,

D D (−∆ + V − z)uD = (−∆ − z)uD 0 + V u0 − (−∆ + V − z) HΩ − zIΩ

−1

V uD 0

D = V uD 0 − IΩ V u0 = 0, N N (−∆ + V − z)uN = (−∆ − z)uN 0 + V u0 − (−∆ + V − z) HΩ − zIΩ N = V uN 0 − IΩ V u0 = 0,

(3.36)

(3.38) −1

V uN 0 (3.39)

3.15 3.16

VARIATIONS ON A THEME OF JOST AND PAIS

2.4

2.5

3.15

3.16

and by (2.4), (2.5) and (3.34), (3.35) one also obtains, −1 D D γD uD = γD uD V u0 0 − γD HΩ − zIΩ −1  −1 −1 D D D = f − γD H0,Ω − zIΩ IΩ + V H0,Ω − zIΩ V u0 = f,  −1 γ eN uN = γ eN uN eN HΩN − zIΩ V uN 0 −γ 0   −1 −1 N −1 N N = g − γN H0,Ω − zIΩ IΩ + V H0,Ω − zIΩ V u0 = g. 3.11

13

3.12

3.3

3.4

3.13

(3.40)

(3.41)

3.14

Finally, (3.30) and (3.31) follow from (3.9), (3.10), (3.32), (3.33), and the resolvent identity,  −1  −1 ∗ ∗ D uD (z) = IΩ − HΩD − zIΩ V − γN H0,Ω − zIΩ f        −1 ∗ −1 ∗ ∗ D = − γN H0,Ω − zIΩ IΩ − HΩD − zIΩ V f     −1 ∗ ∗ D = − γN HΩ − zIΩ f, (3.42)       −1 −1 ∗ ∗ N uN (z) = IΩ − HΩN − zIΩ V γD H0,Ω − zIΩ g     −1 ∗ ∗ −1 ∗ N N = γD H0,Ω − zIΩ IΩ − HΩ − zIΩ V g     −1 ∗ ∗ = γD HΩN − zIΩ g. (3.43) Analytic z ∈ /  continuation with respect to z then permits one to remove the additional condition  D N σ H0,Ω in the case of the Dirichlet problem, and the additional condition z ∈ / σ H0,Ω in the context of the Neumann problem.  h2.1

D (z) associated Assuming Hypothesis 2.1, we now introduce the Dirichlet-to-Neumann map M0,Ω with (−∆ − z) on Ω, as follows, (   H 1 (∂Ω) → L2 (∂Ω; dn−1 σ), D D M0,Ω (z) : z ∈ C σ H0,Ω , (3.44) D f 7→ −e γN u0 ,

3.20

where uD 0 is the unique solution of (−∆ − z)uD 0 = 0 on Ω,

3/2 uD (Ω), 0 ∈H

γD u D 0 = f on ∂Ω,

(3.45)

h3.2

Similarly, assuming Hypothesis 3.2, we introduce the Dirichlet-to-Neumann map MΩD (z), associated with (−∆ + V − z) on Ω, by (   H 1 (∂Ω) → L2 (∂Ω; dn−1 σ), MΩD (z) : z ∈ C σ HΩD , (3.46) D f 7→ −e γN u ,

3.22

where uD is the unique solution of (−∆ + V − z)uD = 0 on Ω, t3.1

uD ∈ H 3/2 (Ω),

γD uD = f on ∂Ω.

(3.47)

t3.3

By Theorems 3.1 and 3.3 one obtains  D M0,Ω (z), MΩD (z) ∈ B H 1 (∂Ω), L2 (∂Ω; dn−1 σ) .

(3.48)

h2.1

N In addition, assuming Hypothesis 2.1, we introduce the Neumann-to-Dirichlet map M0,Ω (z) associated with (−∆ − z) on Ω, as follows, (   L2 (∂Ω; dn−1 σ) → H 1 (∂Ω), N N M0,Ω (z) : z ∈ C σ H0,Ω , (3.49) N g 7→ γD u0 ,

where uN 0 is the unique solution of (−∆ − z)uN 0 = 0 on Ω,

3/2 uN (Ω), 0 ∈H

γ eN uN 0 = g on ∂Ω.

(3.50)

3.24

14

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

h3.2

Similarly, assuming Hypothesis 3.2, we introduce the Neumann-to-Dirichlet map MΩN (z) associated with (−∆ + V − z) on Ω by (   L2 (∂Ω; dn−1 σ) → H 1 (∂Ω), N MΩ (z) : z ∈ C σ HΩN , (3.51) N g 7→ γD u ,

3.26

where uN is the unique solution of (−∆ + V − z)uN = 0 on Ω, t3.1

uN ∈ H 3/2 (Ω),

t3.3

γ eN uN = g on ∂Ω.

(3.52)

Again, by Theorems 3.1 and 3.3 one obtains  N M0,Ω (z), MΩN (z) ∈ B L2 (∂Ω; dn−1 σ), H 1 (∂Ω) .

(3.53)

h2.1

D N (z) and M0,Ω (z), and under the Moreover, under the assumption of Hypothesis 2.1 for M0,Ω h3.2 D N assumption of Hypothesis 3.2 for MΩ (z) and MΩ (z), one infers the following equalities:    N D D N M0,Ω (z) = −M0,Ω (z)−1 , z ∈ C σ H0,Ω ∪ σ H0,Ω , (3.54)    N D −1 D N MΩ (z) = −MΩ (z) , z ∈ C σ HΩ ∪ σ HΩ , (3.55)

3.28 3.29

and  −1 ∗ ∗   D D D M0,Ω (z) = γ eN γN H0,Ω − zIΩ , z ∈ C σ H0,Ω ,       −1 ∗ ∗ MΩD (z) = γ eN γN HΩD − zIΩ , z ∈ C σ HΩD ,       −1 ∗ ∗ N N N M0,Ω (z) = γD γD H0,Ω − zIΩ , z ∈ C σ H0,Ω ,       −1 ∗ ∗ MΩN (z) = γD γD HΩN − zIΩ , z ∈ C σ HΩN . 3.30

(3.56)

3.30

(3.57)

3.31

(3.58)

3.32

(3.59)

3.33

3.33

The representations (3.56)–(3.59) provide a convenient point of departure for proving the operatorWe will return to this topic in a future paper. valued Herglotz property of MΩD and MΩN . 3.30 3.33 Next, we note that the above formulas (3.56)–(3.59) may be used as alternative definitions of the 3.31 Dirichlet-to-Neumann and Neumann-to-Dirichlet maps. In particular, we will next use ( 3.57) and 3.33 (3.59) to extend the above definition of the operators MΩD (z) and MΩN (z) to a more general setting. This is done in the following two lemmas. h2.6

l3.4

Lemma 3.4. Assume Hypothesis 2.6. Then the following boundedness properties hold: −1    γN HΩD − zIΩ ∈ B L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , z ∈ C σ HΩD , −1    γD HΩN − zIΩ ∈ B L2 (Ω; dn x), H 1 (∂Ω) , z ∈ C σ HΩN ,  −1 ∗ ∗    γN HΩD − zIΩ ∈ B H 1 (∂Ω), H 3/2 (Ω) , z ∈ C σ HΩD ,  −1 ∗ ∗    γD HΩN − zIΩ ∈ B L2 (∂Ω; dn−1 σ), H 3/2 (Ω) , z ∈ C σ HΩN . 3.31

(3.60)

3.38a

(3.61)

3.39a

(3.62)

3.40a

(3.63)

3.41a

3.33

Moreover, the operators MΩD (z) in (3.57) and MΩN (z) in (3.59) remain well-defined and satisfy    MΩD (z) ∈ B H 1 (∂Ω), L2 (∂Ω; dn−1 σ) , z ∈ C σ HΩD , (3.64)    N 2 n−1 1 N MΩ (z) ∈ B L (∂Ω; d σ), H (∂Ω) , z ∈ C σ HΩ . (3.65)   In particular, MΩN (z), z ∈ C σ HΩN , are compact operators in L2 (∂Ω; dn−1 σ).    D Proof. We temporarily assume that z ∈ C σ H0,Ω ∪σ HΩD in the case of the Dirichlet Laplacian    N and that z ∈ C σ H0,Ω ∪ σ HΩN in the context of the Neumann Laplacian. Next, let u, v and u e, ve denote the following factorizations of the perturbation V , V (x) = u(x)v(x),

u(x) = exp(i arg(V (x)))|V (x)|1/2 ,

v(x) = |V (x)|1/2 ,

(3.66)

3.42a 3.43a

3.44a

VARIATIONS ON A THEME OF JOST AND PAIS

u e(x) = exp(i arg(V (x)))|V (x)|p/p1 ,

15

ve(x) = |V (x)|p/p2 ,

(3.67)

3.45a

(3.68)

3.46a

(3.69)

3.46b

It follows from the definition of the operators and and, in particular, from (B.5) that h −1 i−1 −1 −1 −1 −1 D D D − zI D − zI v HΩD − zIΩ v IΩ + u H0,Ω u H0,Ω − zIΩ = H0,Ω − zIΩ − H0,Ω Ω Ω −1 i−1 −1 −1 h −1 D D D − zI D − zI e H0,Ω ve u e H0,Ω − zIΩ , ve IΩ + u = H0,Ω − zIΩ − H0,Ω Ω Ω (3.70) h i−1     −1 −1 −1 −1 −1 N N N − zI N − zI HΩN − zIΩ = H0,Ω − zIΩ − H0,Ω v IΩ + u H0,Ω v u H0,Ω − zIΩ Ω Ω −1 h −1 i−1 −1 −1 N N N − zI N − zI ve IΩ + u e H0,Ω ve u e H0,Ω − zIΩ . = H0,Ω − zIΩ − H0,Ω Ω Ω (3.71)

3.47a

V (x) = u e(x)e v (x), where

( 3p/2, p1 = 4p/3, h2.6

3.44a

( 3p, p2 = 4p,

n = 2, n ≥ 3,

n = 2, n ≥ 3.

3.45a

We note that Hypothesis 2.6 and (3.66), (3.67) imply u e ∈ Lp1 (Ω; dn x), ve ∈ Lp2 (Ω; dn x), and u, v ∈ L2p (Ω; dn x). HΩD

B.5

HΩN

3.38a

Next, we establish a number of boundedness properties that will imply (3.60)–(3.65). First, note 2.6 3.46a that it follows from Hypothesis 2.6 and (3.68) that p1 = 32 p > 2 > 2n/3, p2 = 3p > 4 for n = 2 and l2.3 p1 = 43 p > 2n/3, p2 = 4p > 2n for n ≥ 3. Then, utilizing Lemma 2.3, one obtains −1    D D u e H0,Ω − zIΩ ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω , (3.72)     −1 N N u e H0,Ω − zIΩ ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω , (3.73) 1−ε     − 4 D D − zI H0,Ω ve ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω , (3.74) Ω − 1−ε    N 4 N − zI H0,Ω ve ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω , (3.75) Ω l2.2

3.48a

3.43a

3.51a 3.52a 3.53a 3.54a

incl-xxx

and, utilizing Lemma 2.2 and the inclusion (A.4), one obtains for ε ∈ (0, 1 − 2n/p2 ), − 3+ε    3+ε D D 4 H0,Ω − zIΩ , ∈ B L2 (Ω; dn x), H 2 (Ω) , z ∈ C σ H0,Ω 3+ε     3+ε − N N 4 H0,Ω − zIΩ ∈ B L2 (Ω; dn x), H 2 (Ω) , z ∈ C σ H0,Ω .

(3.76)

3.54b

(3.77)

3.54c

(3.78)

3.55a

(3.79)

3.56a

(3.80)

3.57a

(3.81)

3.58a

In addition, D H0,Ω − zIΩ N H0,Ω − zIΩ

− 3+ε 4 − 3+ε 4

: L2 (Ω; dn x) → H

3+ε 2

(Ω) ,→ H 3/2 (Ω),

: L2 (Ω; dn x) → H

3+ε 2

(Ω) ,→ H 3/2 (Ω),

3.53a

3.56a

In particular, one concludes from (3.74)–(3.79) that −1  D − zI H0,Ω ve ∈ B L2 (Ω; dn x), H 3/2 (Ω) , Ω −1  N − zI H0,Ω ve ∈ B L2 (Ω; dn x), H 3/2 (Ω) , Ω 3.53a

3.56a

  D z ∈ C σ H0,Ω ,   N z ∈ C σ H0,Ω .

  D z ∈ C σ H0,Ω ,   N z ∈ C σ H0,Ω .

2.3

incl-xxx

lA.6

In addition, it follows from (3.74)–(3.79), the definition of γN (2.3), inclusion (A.4), and Lemma A.6 that −1    D D − zI γN H0,Ω ve ∈ B L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , z ∈ C σ H0,Ω , (3.82) Ω

3.59a

16

F. GESZTESY, M. MITREA, AND M. ZINCHENKO N − zI γD H0,Ω Ω

−1

 ve ∈ B L2 (Ω; dn x), H 1 (∂Ω) ,

 N z ∈ C\σ(H0,Ω .

(3.83)

3.60a

(3.84)

3.61a

(3.85)

3.62a

t3.1

Next, it follows from Theorem 3.1 that  −1 ∗    D D γN H0,Ω − zIΩ ∈ B H 1 (∂Ω), H 3/2 (Ω) , z ∈ C σ H0,Ω ,       −1 ∗ N N ∈ B L2 (∂Ω; dn−1 σ), H 3/2 (Ω) , z ∈ C σ H0,Ω γD H0,Ω − zIΩ . Then, employing the Sobolev embedding theorem H 3/2 (Ω) ,→ Lq (Ω; dn x)

(3.86)

with q satisfying 1/q = (1/2) − (1/p1 ) > (1/2) − 3/(2n),3.61a n ≥ 2, and3.62a the fact that u e ∈ Lp1 (Ω; dn x), one obtains the following boundedness properties from (3.84) and (3.85),  −1 ∗    D D u e γN H0,Ω ∈ B H 1 (∂Ω), L2 (Ω; dn x) , z ∈ C σ H0,Ω , (3.87) − zIΩ       −1 ∗ N N u e γD H0,Ω − zIΩ ∈ B L2 (∂Ω; dn−1 σ), L2 (Ω; dn x) , z ∈ C σ H0,Ω . (3.88) h i h −1 tB.3 D − zI Moreover, it follows from Theorem B.3 that the operators IΩ + u e H0,Ω ve and IΩ + Ω   −1 i  D N − zI ∪ σ HΩD and u e H0,Ω ve are boundedly invertible on L2 (Ω; dn x) for z ∈ C σ H0,Ω Ω    N z ∈ C σ H0,Ω ∪ σ HΩN , respectively, that is, the following operators are bounded, h −1 i−1     D D − zI IΩ + u e H0,Ω ve ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω ∪ σ HΩD , (3.89) Ω i h −1 −1     N N − zI e H0,Ω ve ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω ∪ σ HΩN . (3.90) IΩ + u Ω 3.47a

3.68a

l3.4

3.47a

3.31

3.43a

3.48a

3.33

3.66a

3.67a 3.68a

3.38a

Finally, combining (3.70)–(3.90), one obtains the assertions of Lemma 3.4 as follows: (3.60) follows 3.47a 3.51a 3.59a 3.67a 3.39a 3.48a 3.52a 3.60a 3.68a 3.40a from (3.47a 3.70), (3.57a 3.72), (3.65a 3.82), (3.67a 3.89); 3.41a (3.61) follows from 3.48a (3.71),3.58a (3.73),3.66a (3.83),3.68a (3.90); (3.62) follows from (3.70), (3.20 3.80), (3.59a 3.87), (3.65a 3.89); (3.63)3.67a follows from (3.71), (3.81), (3.88), (3.90); Thus, by (3.44), (3.82), (3.87), and (3.89), we may introduce the operator −1 h −1 i−1 −1 ∗ D D D − zI D − zI MΩD (z) = M0,Ω (z)−γN H0,Ω ve IΩ + u e H0,Ω ve , (3.91) u e γN H0,Ω −zIΩ Ω Ω 3.42a

3.65a

and observeh2.6 that it satisfies (3.64). In addition, (3.70) shows that (3.57) remains in effect under Hypothesis 2.6. 3.24 3.60a 3.66a 3.68a Similarly, by (3.49), (3.83), (3.88), and (3.90), we may introduce the operator −1 h −1 i−1 −1 ∗ N N N − zI N − zI MΩN (z) = M0,Ω (z)−γD H0,Ω ve IΩ + u e H0,Ω ve u e γD H0,Ω −zIΩ , (3.92) Ω Ω

3.49a

3.50a

and observeh2.6 that it satisfies (3.65). In addition, (3.71) shows that (3.59) remains inEQ1 effect under MM07 Hypothesis 2.6. Moreover, since H 1 (∂Ω) embeds compactly into L2 (∂Ω; dn−1 σ) (cf. (A.6) and [55, Proposition 2.4]), MΩN3.31 (z), z ∈ C 3.33 σ HΩN , are compact operators in L2 (∂Ω; dn−1 σ). Finally, formulas (3.57) and (3.59) together with analytic  continuationNwith  respect to z then D permit one to remove the additional restrictions z ∈ / σ H0,Ω and z ∈ / σ H0,Ω , respectively.  Actually, one can go a step further and allow an additional perturbation V1 ∈ L∞ (Ω; dn x) of HΩD and HΩN , D H1,Ω = HΩD + V1 ,

D dom(H1,Ω ) = dom(HΩD ),

(3.93)

3.70a

N H1,Ω

N dom(H1,Ω )

(3.94)

3.70b

=

HΩN

+ V1 ,

=

dom(HΩN ).

VARIATIONS ON A THEME OF JOST AND PAIS

17

D N Defining the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators M1,Ω and M1,Ω in an anal3.31 3.33 ogous fashion as in (3.57) and (3.59),    −1 ∗ ∗ D D D M1,Ω (z) = γ eN γN H1,Ω − zIΩ , z ∈ C σ H1,Ω , (3.95)  −1 ∗ ∗   N N N M1,Ω (z) = γD γD H1,Ω − zIΩ , z ∈ C σ H1,Ω , (3.96)

3.71a 3.72a

one can then prove the following result: h2.6

D Lemma 3.5. Assume Hypothesis 2.6 and let V1 ∈ L∞ (Ω; dn x). Then the operators M1,Ω (z) and 3.71a 3.72a N M1,Ω (z) defined by (3.95) and (3.96) satisfy the following boundedness properties,    D D M1,Ω , (3.97) (z) ∈ B H 1 (∂Ω), L2 (∂Ω; dn−1 σ) , z ∈ C σ H1,Ω    N 2 n−1 1 N M1,Ω (z) ∈ B L (∂Ω; d σ), H (∂Ω) , z ∈ C σ H1,Ω . (3.98)    D D Proof. We temporarily assume that z ∈ C σ HΩD ∪ σ H1,Ω in the case of M1,Ω and that z ∈    N N N C σ HΩ ∪ σ H1,Ω in the context of M1,Ω . 3.70a 3.70b Next, using resolvent identities and (3.93), (3.94), one computes −1 −1 −1 h −1 i−1 −1 D H1,Ω − zIΩ = HΩD − zIΩ − HΩD − zIΩ IΩ + V1 HΩD − zIΩ V1 HΩD − zIΩ , (3.99) h i−1     −1 −1 −1 −1 −1 N H1,Ω − zIΩ = HΩN − zIΩ − HΩN − zIΩ IΩ + V1 HΩN − zIΩ V1 HΩN − zIΩ , (3.100)

3.73a 3.74a

3.75a

3.76a

and hence, −1 i−1  −1 ∗ ∗ IΩ + V1 HΩD − zIΩ V1 γN HΩD − zIΩ , −1 h −1 i−1  −1 ∗ ∗ − zIΩ IΩ + V1 HΩN − zIΩ V1 γD HΩN − zIΩ .

D M1,Ω = MΩD − γN HΩD − zIΩ N M1,Ω = MΩN − γD HΩN

3.73a

−1 h

3.74a

3.38a

3.43a

(3.101)

3.77a

(3.102)

3.78a

tB.3

The assertions (3.97) and (3.98) now follow from (3.60)–(3.65) and the fact that by Theorem B.3, the −1  −1    N are boundedly invertible on L2 (Ω; dn x) and IΩ +V1 H operators IΩ+V1 HΩD−zIΩ Ω −zIΩ    3.71a D D N N for all z ∈ C σ HΩ ∪ σ H1,Ω and z ∈ C σ HΩ ∪ σ H1,Ω , respectively. Formulas (3.95) 3.72a and (3.96) together with analytic respect to z then permit one to remove the   continuationNwith  / σ HΩ , respectively. additional restrictions z ∈ / σ HΩD and z ∈ Weyl–Titchmarsh operators, in a spirit close to ours, have recently been discussed by Amrein and AP04 Pearson [2] in connection with the interior and exterior of a ball in R3 and real-valued potentials V ∈ L∞ (R3 ; d3 x). For additional literature on Weyl–Titchmarsh operators,ABMN05 relevant inBMN00 theBMN02 context BL06BMN06 BM04 of boundary value spaces (boundary triples, etc.), we refer, for instance, to [1], [5], [6], [9], [10], [11], DM91 DM95 GKMT01GG91 MM06 Ma04 MPP07Pa87 Pa02 [21], [22], [30], [36, Ch. 3], [51], [52], [54] [70], [71]. For applicationsCh90 of the Dirichlet-to-Neumann NSU88 PS02 Sa05 SU86 SU87 map to Borg–Levinson-type inverse spectral problems we refer to [17], [59], [67], [77], [85], [86] KLW05 (see also [49] for an alternative approach based on the boundary control method). The inverse problem of detecting the number of connected components (i.e., the number of holes) in ∂Ω using HL01 the high-energy spectral asymptotics of the Dirichlet-to-Neumann map is studied in [40], t4.2 Next, we prove the following auxiliary result, which will play a crucial role in Theorem 4.3, the principal result of this paper. h2.6

l3.5

Lemma 3.6. Assume Hypothesis 2.6. Then the following identities hold, −1  −1 ∗ ∗ D D − zI M0,Ω (z) − MΩD (z) = γ eN HΩD − zIΩ V γN H0,Ω , Ω    D D z ∈ C σ H0,Ω ∪ σ HΩ ,

(3.103)

3.35

18

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

−1  −1 ∗ ∗ D N − zI MΩD (z)M0,Ω (z)−1 = I∂Ω − γ eN HΩD − zIΩ V γD H0,Ω , Ω     D D N z ∈ C σ H0,Ω ∪ σ HΩ ∪ σ H0,Ω . (3.104)    3.35 3.30 3.31 D Proof. Let z ∈ C σ H0,Ω ∪ σ HΩD . Then (3.103) follows from (3.56), (3.57), and the resolvent identity  −1 −1 ∗ ∗ D D M0,Ω (z) − MΩD (z) = γ eN γN H0,Ω − zIΩ − HΩD − zIΩ  −1 −1 ∗ ∗ D − zI eN γN HΩD − zIΩ =γ V H0,Ω (3.105) Ω      −1 −1 ∗ ∗ D − zI eN HΩD − zIΩ V γN H0,Ω =γ . Ω     3.28 3.32 3.35 D N Next, if z ∈ C σ H0,Ω ∪ σ HΩD ∪ σ H0,Ω , then it follows from (3.54), (3.58), and (3.103) that  D D D MΩD (z)M0,Ω (z)−1 = I∂Ω + MΩD (z) − M0,Ω (z) M0,Ω (z)−1  D N = I∂Ω + M0,Ω (z) − MΩD (z) M0,Ω (z)   −1 ∗ ∗ −1 D − zI = I∂Ω + γ eN HΩD − zIΩ V γN H0,Ω (3.106) Ω     −1 ∗ ∗ N × γD γD H0,Ω − zIΩ .

3.36

3.40

t3.1

Let g ∈ L2 (∂Ω; dn−1 σ). Then by Theorem 3.1,  −1 ∗ ∗ N u = γD H0,Ω − zIΩ g

(3.107)

3.41

is the unique solution of (−∆ − z)u = 0 on Ω,

u ∈ H 3/2 (Ω),

γ eN u = g on ∂Ω.

(3.108)

t3.1

Setting f = γD u ∈ H 1 (∂Ω) and utilizing Theorem 3.1 once again, one obtains −1 ∗  D f u = − γN H0,Ω − zIΩ  −1 ∗ ∗  −1 ∗ ∗ D N = − γN H0,Ω − zIΩ γD γD H0,Ω − zIΩ g. 3.41

Thus, it follows from (3.107) and (3.109) that  −1 ∗ ∗  −1 ∗ ∗  −1 ∗ ∗ D N N γN H0,Ω − zIΩ γD γD H0,Ω − zIΩ = − γD H0,Ω − zIΩ . 3.44

(3.109)

3.43

(3.110)

3.44

3.43

3.40

3.36

Finally, insertion of (3.110) into (3.106) yields (3.104). 4.24

4.29a

 3.35

It follows from (4.38)–(4.44) that γ eN can be replaced by γN on the right-hand side of (3.103) and 3.36 (3.104). 3.36 We note that the right-hand side (and hence the left-handside) of (3.104) permits an analytic   D N continuation to z ∈ σ H0,Ω as long as z ∈ / σ HΩD ∪ σ H0,Ω . s4

4. A Multi-Dimensional Variant of a Formula due to Jost and Pais In this section we prove our multi-dimensional variants of the Jost and Pais formula as discussed in the introduction. We start with an elementary comment on determinants which, however, lies at the heart of the matter of our multi-dimensional variant of the one-dimensional Jost and Pais result. Suppose A ∈ B(H1 , H2 ), B ∈ B(H2 , H1 ) with AB ∈ B1 (H2 ) and BA ∈ B1 (H1 ). Then, det(IH2 − AB) = det(IH1 − BA).

(4.1)

4.0

VARIATIONS ON A THEME OF JOST AND PAIS

19

4.0

Equation (4.1) follows from the fact that all nonzero eigenvalues of AB and BA coincide including their algebraic multiplicities. The latter fact, in turn, can be derived from the formula A(BA − zIH1 )−1 B = IH2 + z(AB − zIH2 )−1 ,

z ∈ C\(σ(AB) ∪ σ(BA))

(4.2)

De78

(and its companion with A and B interchanged), as discussed in detail by Deift [19]. In particular, H1 and H2 may have different dimensions. Especially, one of them may be infinite 4.0 and the other finite, in which case one of the two determinants in (4.1) reduces to a finite determinant. JP51 [43] as described This case indeed occurs in the original one-dimensional case studied by Jost and Pais GM03 t4.1 in detail in [32] and the references therein. In the proof of Theorem 4.2 below, the role of H1 and H2 will be played by L2 (Ω; dn x) and L2 (∂Ω; dn−1 σ), respectively. In the context of KdV flows and reflectionless (i.e., generalizations of soliton-type) potentials represented as Fredholm determinants, a reduction of such determinants (in some cases to finite determinants) has also been studied by Ko04 Kotani [48], relying on certain connections to stochastic analysis. We start with an auxiliary lemma which is of independent interest in the area of modified Fredholm determinants. l4.1

Lemma 4.1. Let H be a separable, complex Hilbert space, and assume A, B ∈ Bk (H) for some fixed k ∈ N. Then there exists a polynomial Tk (·, ·) in A and B with Tk (A, B) ∈ B1 (H), such that the following formula holds det k ((IH − A)(IH − B)) = det k (IH − A) det k (IH − B)etr(Tk (A,B)) .

(4.3)

4.3a

Moreover, Tk (·, ·) is unique up to cyclic permutations of its terms, and an explicit formula for Tk may be derived from the representation Tk (A, B) =

2k−2 X

Pm (A, B),

(4.4)

4.4a

m=k

where Pm (·, ·), m = 1, . . . , 2k − 2, denote homogeneous polynomials in A and B of degree m (i.e., each term of Pm (A, B) contains precisely the total number m of A’s and B’s) that one obtains after rearranging the following expression in powers of t, k−1 X j=1

X  2k−2 1 tm Pm (A, B), (tA + tB − t2 AB)j − (tA)j − (tB)j = j m=1 4.4a

t ∈ R.

(4.5)

4.5a

4.5a

In particular, computing Tk (A, B) from (4.4) and (4.5), and subsequently using cyclic permutations 4.3a to simplify the resulting expressions, then yields for the terms Tk (A, B) in (4.3) T1 (A, B) = 0, T2 (A, B) = − AB, 1 T3 (A, B) = − A2 B − AB 2 + ABAB, 2 1 1 3 3 T4 (A, B) = − A B − AB − ABAB − A2 B 2 + A2 BAB + AB 2 AB − ABABAB, (4.6) 2 3 T5 (A, B) = − A4 B − AB 4 − A3 B 2 − A2 B 3 − A2 BAB − AB 2 AB + A3 BAB + AB 3 AB 2 1 1 + A2 B 2 AB + A2 BAB 2 + ABABAB + A2 BA2 B + AB 2 AB 2 3 2 2 1 2 2 − A BABAB − AB ABAB + ABABABAB, etc. 4 Si05 Proof. Suppose temporarily that A, B ∈ B1 (H). Then it follows from [82, Theorem 9.2] that det k ((IH − A)(IH − B)) = det k (IH − A) det k (IH − B)etr(Tk (A,B)) , e

(4.7)

4.6a

20

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

where Tek (A, B) =

k−1 X j=1

 1 (A + B − AB)j − (A)j − (B)j , j

(4.8)

4.5a

and hence, by (4.5)

Tek (A, B) =

2k−2 X

Pm (A, B).

(4.9)

4.9a

m=1

Since tr(·) is linear and invariant under cyclic permutation of its argument, it remains to show that 4.4a 4.9a Tk (A, B) in (4.4) and Tek (A, B) in (4.9) are equal up to cyclic permutations of their terms, that is, to show that Pm (A, B) vanish for m = 1, . . . , k − 1 after a finite number of cyclic permutations of their terms. Let Pem (·, ·), m ≥ 1, denote a sequence of polynomials in A and B, obtained after rearranging the following expression in powers of t ∈ C, ln((IH − tA)(IH − tB)) − ln(IH − tA) − ln(IH − tB) =

∞ X 1 j=1

j

∞  X (tA + tB − t2 AB)j − (tA)j − (tB)j = tm Pem (A, B) for |t| sufficiently small. m=1

(4.10) 4.5a

4.10a

4.10a

Then it follows from (4.5) and (4.10) that Pm (A, B) = Pem (A, B) for m = 1, . . . , k − 1, and hence, it suffices to show that Pem (A, B) vanish for m = 1, . . . , k−1 after a finite number of cyclic permutations of their terms. The latter fact now follows from the Baker–Campbell–Hausdorff (BCH) formula as follows: First, assume D, E ∈ B(H), H. Then, etD etE = etD+tE+F (t) for |t| sufficiently small,

(4.11)

where F (t) is given by a norm convergent infinite sum of certain repeated commutators involving D Su77 BC04 and E, as discussed, for instance, in [84] (cf. also [7]). Explicitly, F is of the form   X ∞ ∞ X ∞ j+k X 1 dp t ` j k , p ∈ N, p ≥ 2, t F` , F p = F (t) = D E (4.12) ln p p! dt j!k! t=0 j=0 `=2

k=0

where

1 1 1 [D, E], F3 = [F2 , E − D], F4 = [[F2 , D], E], etc. (4.13) 2 6 12 That each F` , ` ≥ 2, Bo89 is indeed at mostOt91 a finite sum of commutators follows from a formula derived by Dynkin (cf., e.g., [8, eqs. (1)–(4)], [66, eqs. (2.5), (2.6), (3.7), (3.8)]). If in addition, D, E ∈ B1 (H), the expression for F (t) is actually convergent in the B1 (H)-norm for |t| sufficiently small. Thus, F (t) vanishes after a finite number of cyclic permutations of each of its coefficients Fn . 4.12a Next, setting D = ln(IH − tA), E = ln(IH − tB) and taking the natural logarithm in (4.11) then implies ln((IH − tA)(IH − tB)) − ln(IH − tA) − ln(IH − tB) = F (t) (4.14) and hence ln((IH − tA)(IH − tB)) − ln(IH − tA) − ln(IH − tB) = 0 (4.15) P∞ ` after a finite number of cyclic permutations in each of the coefficients F` in F (t) = `=2 t F` . Thus, 4.10a by (4.10), each Pem (A, B), m ≥ 1, vanishes after a finite number of cyclic permutations of its terms. Consequently, Pm (A, B) vanish for m = 1, . . . , k − 1 after a finite number of cyclic permutations of their terms. F2 =

4.12a

VARIATIONS ON A THEME OF JOST AND PAIS

21

Finally, to remove the assumption A, B ∈ B1 (H), one uses a standard approximation argument 4.3a of operators in Bk (H) by operators in B1 (H), together with the fact that both sides of (4.3) are well-defined for A, B ∈ Bk (H).  GLMZ05

t4.1

Next, we prove an extension of a result in [31] to arbitrary space dimensions:     h2.6 D N Theorem 4.2. Assume Hypothesis 2.6, let k ∈ N, k ≥ p, and z ∈ C σ HΩD ∪σ H0,Ω ∪σ H0,Ω . Then, −1  −1 ∗   N − zI γN HΩD − zIΩ V γD H0,Ω ∈ Bp L2 (∂Ω; dn−1 σ) ⊂ Bk L2 (∂Ω; dn−1 σ) (4.16) Ω

4.2

and  −1  N − zI det k IΩ + u H0,Ω v Ω    −1 D − zI v det k IΩ + u H0,Ω Ω  −1  −1 ∗   N − zI = det k I∂Ω − γN HΩD − zIΩ V γD H0,Ω exp tr(Tk (z)) . (4.17) Ω  2 Here Tk (z) ∈ B1 Ll4.1 (∂Ω; dn−1 σ) denotes one of the cyclic permutations of the polynomial Tk (·, ·) defined in Lemma 4.1 with the following choice of A = A0 (z) and B = B0 (z), with A0 (z) and B0 (z) given by h −1 i∗ −1   N − zI u e γN HΩD − zIΩ ve ∈ Bp L2 (Ω; dn x) ⊂ Bk L2 (Ω; dn x) , A0 (z) = γD H0,Ω Ω (4.18) −1   D − zI B0 (z) = −e u H0,Ω ve ∈ Bp L2 (Ω; dn x) ⊂ Bk L2 (Ω; dn x) , Ω

4.3

4.4

and the functions u, v, u e, and ve are given by u(x) = exp(i arg(V (x)))|V (x)|1/2 , p/p1

u e(x) = exp(i arg(V (x)))|V (x)|

v(x) = |V (x)|1/2 , ,

p/p2

ve(x) = |V (x)|

(4.19) ,

(4.20)

with ( 3p/2, p1 = 4p/3,

n = 2, n ≥ 3,

( 3p, p2 = 4p,

n = 2, n ≥ 3,

(4.21)

4.6

and V = uv = u ˜v˜. In particular, −1 −1 h −1 i∗  D − zI N − zI T2 (z) = γN H0,Ω V HΩD − zIΩ V γD H0,Ω ∈ B1 L2 (∂Ω; dn−1 σ) . (4.22) Ω Ω

4.5

4.3

2.35

Proof. From the that side of (4.17) is well-defined by (2.32). Let  outset we note  the left-hand D N z ∈ C σ HΩD ∪σ H0,Ω ∪σ H0,Ω and note that p11 + p12 = p1 for all n ≥ 2, and hence V = uv = u eve. Next, we introduce −1 −1 D − zI N − zI KD (z) = −u H0,Ω v, KN (z) = −u H0,Ω v (4.23) Ω Ω B.4

4.7

tB.3

(cf. (B.4)) and note that by Theorem B.3

 [IΩ − KD (z)]−1 ∈ B L2 (Ω; dn x) ,

z∈C



  D . σ HΩD ∪ σ H0,Ω

(4.24)

l4.1

e0 (z) and B = B e0 (z) defined by Then Lemma 4.1 with A = A e0 (z) = IΩ − (IΩ − KN (z))[IΩ − KD (z)]−1 = (KN (z) − KD (z))[IΩ − KD (z)]−1 , A  e0 (z) = KD (z) = −u H D − zIΩ −1 v, B 0,Ω

(4.25)

4.10

(4.26)

4.10A

22

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

yields  −1   N − zI det k IΩ + u H0,Ω v Ω det k IΩ − KN (z)  −1  = det I − K (z) D − zI k Ω D det k IΩ + u H0,Ω v Ω   e0 (z), B e0 (z))) , = det k IΩ − (KN (z) − KD (z))[IΩ − KD (z)]−1 exp tr(Tk (A

(4.27)

4.12

4.4a

where T (·, ·) is the polynomial defined in (4.4). Explicit formulas for the first few Tk are computed 4.6a k in (4.6). lA.3 ∞ Next, temporarily suppose that V ∈ Lp (Ω; dn x) ∩ L (Ω; dnNa1-bis x). Using Lemma A.3 (an extension Na01 rA.5 of a result of Nakamura [60, Lemma 6]) and Remark A.5 (cf. (A.29)), one finds  D −1 −1  N − zI KN (z) − KD (z) = u H0,Ω − zIΩ − H0,Ω v Ω    −1 −1 ∗ N − zI D − zI (4.28) γN H0,Ω v = u γD H0,Ω Ω Ω i h ∗ −1 −1 N − zI D − zI u γN H0,Ω v. = γD H0,Ω Ω Ω 4.13

4.10

4.13

4.7

Inserting (4.28) into (4.25) and utilizing (4.23) and the following resolvent identity which follows B.5 from (B.5), −1 −1 h −1 i−1 D − zI D − zI HΩD − zIΩ v = H0,Ω v IΩ + u H0,Ω v , (4.29) Ω Ω

4.13a

e0 (z), one obtains the following equality for A h i∗   e0 (z) = γD H N − zIΩ −1 u γN H D − zIΩ −1 v. A 0,Ω Ω

4.4A

4.13

(4.30)

4.12

Moreover, insertion of (4.28) into (4.27) yields  −1  N − zI det k IΩ + u H0,Ω v Ω  −1  D − zI det k IΩ + u H0,Ω v Ω  h −1 h −1 i−1  −1 i∗ N − zI D − zI D − zI = det k IΩ − γD H0,Ω u γN H0,Ω v IΩ + u H0,Ω v (4.31) Ω Ω Ω  e0 (z), B e0 (z))) . × exp tr(Tk (A c2.5

4.6

Utilizing Corollary 2.5 with p1 and p2 as in (4.21), one finds −1  N − zI γD H0,Ω u ∈ Bp1 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , Ω −1  D − zI γN H0,Ω v ∈ Bp2 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , Ω

(4.32) (4.33)

and hence, h −1 i∗ −1   N − zI D − zI γD H0,Ω u γN H0,Ω v ∈ Bp L2 (Ω; dn x) ⊂ Bk L2 (Ω; dn x) , Ω Ω −1 h −1 i∗   D − zI N − zI γN H0,Ω v γD H0,Ω u ∈ Bp L2 (∂Ω; dn−1 σ) ⊂ Bk L2 (∂Ω; dn−1 σ) . Ω Ω

(4.35)

Then, using the fact that h −1 i−1  D − zI IΩ + u H0,Ω v ∈ B L2 (Ω; dn x) , Ω

(4.36)

z∈C



  D σ HΩD ∪ σ H0,Ω ,

(4.34)

4.14

VARIATIONS ON A THEME OF JOST AND PAIS

4.0

23

4.14

one applies the idea expressed in formula (4.1) and rearranges the terms in (4.31) as follows:  −1  N − zI det k IΩ + u H0,Ω v Ω (4.37)  −1  D − zI det k IΩ + u H0,Ω v Ω  −1 i−1 h −1 i∗  −1 h D − zI D − zI N − zI = det k I∂Ω − γN H0,Ω γD H0,Ω u v v IΩ + u H0,Ω Ω Ω Ω  e0 , B e0 )) . × exp tr(Tk (A  e0 (z), B e0 (z) to get an operator Similarly, using the cyclicity property of tr(·), one rearranges Tk A on L2 (∂Ω; dn−1 σ) which in the following we denote by Tk (z). This is always possible since each  4.4 4.10A e0 (z), B e0 (z) has at least one factor of A e0 (z). Then using equalities (4.18), (4.26), term of Tk A 4.4A (4.30), and uv = u ˜v˜, one concludes that Tk (z) is a cyclic permutation of Tk (A0 , B0 ) with A0 (z) and  4.4 e0 (z), B e0 (z) = −A e0 (z)B e0 (z) or equivalently B0 (z) given by (4.18). In particular, rearranging T2 A e e T2 (A0 (z),4.5 B0 (z)) = −A4.3 0 (z)B0 (z), one obtains T2 (z) = −B0 (z)A0 (z) = −B0 (z)A0 (z), and hence equality ( 4.22). Thus, ( 4.17), subject to the extra assumption V ∈ Lp (Ω; dn x) ∩ L∞ (Ω; dn x), follows 4.13a 4.20 from (4.29) and (4.37). tB.3 l2.3 c2.5 Finally, assuming only V ∈ Lp (Ω; dn x) and utilizing Theorem B.3, Lemma 2.3, and Corollary 2.5 once again, one obtains h −1 i−1  D − zI IΩ + u e H0,Ω ve ∈ B L2 (Ω; dn x) , (4.38) Ω −p/p1  D u e H0,Ω − zIΩ ∈ Bp1 L2 (Ω; dn x) , (4.39)   −p/p 2 D ve H0,Ω − zIΩ ∈ Bp2 L2 (Ω; dn x) , (4.40)   −1 N − zI γD H0,Ω u e ∈ Bp1 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , (4.41) Ω   −1 D − zI γN H0,Ω ve ∈ Bp2 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , (4.42) Ω

4.20

4.24 4.25 4.26 4.27 4.28

and thus, 4.24

D − zI u e H0,Ω Ω

4.29

−1

  ve ∈ Bp L2 (Ω; dn x) ⊂ Bk L2 (Ω; dn x) .

(4.43)

Relations (4.38)–(4.43) together with the following resolvent identity that follows from (B.5), −1 −1 h −1 i−1 D − zI D − zI HΩD − zIΩ ve = H0,Ω ve IΩ + u e H0,Ω ve , (4.44) Ω Ω 4.2

4.4

4.5

prove the Bk -property (4.16), (4.18), and (4.22), and hence,2.8 the leftand 2.28 the right-hand sides of 4.3 2.27 (4.17) are well-defined for V ∈ Lp (Ω; dn x). Thus, using (2.9), (2.26), (2.27), the continuity of  , the continuity of tr(·) with respect to the det k (·) with respect to the Bk -norm k · k Bk L2 (Ω;dn x)  , and an approximation of V ∈ Lp (Ω; dn x) by a sequence of potentials trace norm k · k B1 L2 (Ω;dn x)

Vj ∈ Lp (Ω; dn x) ∩ L∞ (Ω; dn x), j ∈ N, in the norm of Lp (Ω; dn x) as j ↑ ∞, then extends the result from V ∈ Lp (Ω; dn x) ∩ L∞ (Ω; dn x) to V ∈ Lp (Ω; dn x). 

t4.2

4.29

B.5

Given these preparations, we are ready for the principal result of this paper, the multi-dimensional t1.2 analog of Theorem 1.2:     h2.6 D N Theorem 4.3. Assume Hypothesis 2.6, let k ∈ N, k ≥ p, and z ∈ C σ HΩD ∪σ H0,Ω ∪σ H0,Ω . Then, −1 h −1 i∗  D N − zI MΩD (z)M0,Ω (z)−1 −I∂Ω = −γN HΩD − zIΩ V γD H0,Ω ∈ Bk L2 (∂Ω; dn−1 σ) (4.45) Ω

4.29a

24

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

and  −1  N − zI det k IΩ + u H0,Ω v Ω  −1  D − zI det k IΩ + u H0,Ω v Ω  −1 ∗   −1  N − zI = det k I∂Ω − γN HΩD − zIΩ exp tr(Tk (z)) V γD H0,Ω Ω   D = det k MΩD (z)M0,Ω (z)−1 exp tr(Tk (z))

(4.46)

4.30

(4.47)

4.31

t4.1

with Tk (z) defined in Theorem 4.2.

l3.5

r4.4

t4.1

Proof. The result follows from combining Lemma 3.6 and Theorem 4.2.      h2.6 N D N Remark 4.4. Assume Hypothesis 2.6, let k ∈ N, k ≥ p, and z ∈ C σ HΩ ∪ σ H0,Ω ∪ σ H0,Ω . Then, −1 h −1 ∗ i∗  N D − zI M0,Ω (z)−1 MΩN (z) − I∂Ω = γN H0,Ω V γD HΩN − zIΩ ∈ Bk L2 (∂Ω; dn−1 σ) Ω (4.48) 4.30 and one can also prove the following analog of (4.46):   D − zI )−1 v det k IΩ + u(H0,Ω Ω   N − zI )−1 v det k IΩ + u(H0,Ω Ω      D − zI )−1 V γ ((H N − zI )−1 )∗ ∗ exp tr(T (z)) , = det k I∂Ω + γN (H0,Ω (4.49) D Ω k Ω Ω   N = det k M0,Ω (z)−1 MΩN (z) exp tr(Tk (z)) , (4.50) where Tk (z) denotes one of the cyclic permutations of the polynomial Tk (A, B) defined in Lemma l4.1 4.1 with the following choice of A = A1 (z) and B = B1 (z), h −1   −1 i∗ D − zI ve ∈ Bp L2 (Ω; dn x) ⊂ Bk L2 (Ω; dn x) , A1 (z) = − γD HΩN − zIΩ u e γN H0,Ω Ω −1   N − zI u H0,Ω B1 (z) = −e ve ∈ Bp L2 (Ω; dn x) ⊂ Bk L2 (Ω; dn x) , Ω and the functions u, v, u e, and ve are given by u(x) = exp(i arg(V (x)))|V (x)|1/2 , p/p1

u e(x) = exp(i arg(V (x)))|V (x)|

v(x) = |V (x)|1/2 , ,

p/p2

ve(x) = |V (x)|

(4.51) ,

(4.52)

with ( 3p/2, p1 = 4p/3,

n = 2, n ≥ 3,

( 3p, p2 = 4p,

n = 2, n ≥ 3,

(4.53)

and V = uv = u ˜v˜. In particular, D − zI T2 (z) = −γN H0,Ω Ω

−1

V HΩN − zIΩ

−1 h −1 i∗ N − zI V γD H0,Ω . Ω

(4.54) t4.2

r4.5

Remark 4.5. It seems tempting at this point BL06 to turn to an abstract version of Theorem 4.3 using DM91 DM95 GG91 the notion of boundary value spaces (see, e.g., [5], [21], [22], [36, Ch. 3] and the references therein).s2 However, the analogs of the necessary mapping and trace ideal properties as recorded in Sections 2 s3 and 3 do not seem to be available at the present time for general self-adjoint extensions of a densely defined, closed symmetric operator (respectively, maximal accretive extensions of closed accretive operators) in a separable complex Hilbert space. For this reason we decided to start with the special, but important case of multi-dimensional Schr¨odinger operators.

4.32

4.33 4.34

VARIATIONS ON A THEME OF JOST AND PAIS

25

A few comments are in order at this point: 4.30 4.31 4.32 The sudden appearance of the exponential term exp(tr(Tk (z))) in (4.46), (4.47), and (4.48), when compared to the one-dimensional case, is due to the necessary use of the modified determinant t4.1 t4.2 detk (·), k ≥ 2, in Theorems 4.2 and 4.3. 4.30 1.16 As mentioned in the introduction, the multi-dimensional extension (4.46) of (1.16), under the GLMZ05 stronger hypothesis V ∈ L2 (Ω; dn x), n = 2, GLMZ05 3, first appeared in [31]. However, the present results in t4.2 Theorem 4.3 go decidedly beyond those in [31] in theh2.1 following sense: (i) the class of domains Ω permitted by Hypothesis 2.1 is substantiallly expanded as compared to GLMZ05 [31]. h2.6 (ii) For n = 2, 3, the conditions on V satisfying Hypothesis 2.6 are now nearly optimal Si71 by comparison Ch84 RS75 [18], Reed and Simon [72, Sect. IX.4], Simon [78, Sect. I.1]). with the Sobolev inequality (cf. Cheney4.31 1.17 (iii) The multi-dimensional extension (4.47) of (1.17) invoking Dirichlet-to-Neumann maps is a new (and the most significant)GLMZ05 result in this paper. (iv) While the results in [31] were confined to dimensions n = 2, 3, all results in this paper are now derived in the general case n ∈ N, n ≥ 2. t4.2 The principal reduction in Theorem 4.3 reduces (a ratio of) 4.30 modified Fredholm determinants associated with operators in L2 (Ω; dn x) on the left-hand side of (4.46) to modified Fredholm deter4.30 2 n−1 4.46) and especially, minants associated with operators in L (∂Ω; d σ) on the right-hand side of ( 4.31 in (4.47). This is the analog of the reduction described in the one-dimensional context of Theot1.2 rem 1.2, where Ω corresponds to the half-line (0, ∞) and its boundary ∂Ω thus corresponds to the one-point set {0}. In the context of elliptic operators on smooth k-dimensional manifolds, the idea of reducing a ratio of zeta-function regularized determinants to a calculation over the k − 1-dimensional boundary Fo87 has been studied by Forman [26]. He also pointed out that if the manifold consists of an interval, the special case of a pair of boundary points then permits one to reduce the zeta-function regularized determinant to the determinant of a finite-dimensional matrix. The latter case is of course an analog oft1.2 the one-dimensional Jost and Pais formula mentioned in the introduction (cf. Theorems t1.1 1.1 and 1.2). Since then, this topic has been further developed in various directions and we refer, BFK91 BFK92 BFK93 BFK95 Ca02 for instance, to Burghelea, Friedlander, and Kappeler [12], [13], [14], [15], Carron [16], Friedlander Fr05 GG07 Mu98 Ok95 Ok95a PW05 [27], Guillarmou and Guillop´ e [39], M¨ u ller [58], Okikiolu [64], [65], Park and Wojciechowski [68], PW05a [69], and the references therein. t4.2

tB.3

4.30

4.33

Combining Theorems 4.3 and B.3 yields the following applications of (4.46) and (4.49): h2.6

t4.6

Theorem 4.6. Assume Hypothesis 2.6 and k ∈ N, k ≥ p. (i)One infers that      D N for all z ∈ C σ HΩD ∪ σ H0,Ω ∪ σ H0,Ω , one has z ∈ σ HΩN  −1  −1 ∗  N − zI if and only if det k I∂Ω − γN HΩD − zIΩ = 0. V γD H0,Ω Ω (ii) Similarly, one infers that      N D for all z ∈ C σ HΩN ∪ σ H0,Ω ∪ σ H0,Ω , one has z ∈ σ HΩD  −1  −1 ∗ ∗  D − zI if and only if det k I∂Ω + γN H0,Ω V γD HΩN − zIΩ = 0. Ω tB.3

(4.55)

4.49

(4.56)

4.50

Proof. By the Birman–Schwinger principle, asdiscussed in Theorem B.3, for any k ∈ N such that    D N ∪ σ H0,Ω , one has k ≥ p and z ∈ C σ HΩD ∪ σ H0,Ω  −1   N − zI z ∈ σ HΩN if and only if det k IΩ + u H0,Ω v = 0. (4.57) Ω

26

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

4.49

4.30

4.50

4.33

Thus, (4.55) follows from (4.46). In the same manner, (4.56) follows from (4.49).



We conclude with another application to eigenvalue counting functions in the case where HΩD and are self-adjoint and have purely discrete spectra (i.e., empty essential spectra). To set the stage we introduce the following assumptions: HΩN

h2.6

h4.7

Hypothesis 4.7. In addition to assuming Hypothesis 2.6 suppose that V is real-valued and that HΩD and HΩN have purely discrete spectra.

r4.8

Remark 4.8. B.11 (i) Real-valuedness of V implies self-adjointness of HΩD and HΩN as noted in (B.11). D N (ii) Since ∂Ω is assumed to be compact, purely discrete spectra of H0,Ω and H0,Ω , that is, comRS78 pactness of their resolvents (cf. [74, Sect. XIII.14]), is equivalent to Ω being bounded. Indeed, if Ω had an unbounded component, then one can construct Weyl sequences which would yield nonempty D N D has empty essential spectrum for . On the other hand, H0,Ω and H0,Ω essential spectra of H0,Ω RS78 n any bounded open set Ω ⊂ R as discussed in the Corollary to [74, Theorem XIII.73]. Similarly, N H0,Ω has empty essential spectrum for any bounded open set Ω satisfying the segment property as RS78 discussed in Corollary 1Gr85 to [74, Theorem XIII.75]. Since any bounded Lipschitz domain satisfies the h2.1 segment property (cf. [37, Sect, 1.2.2]), any bounded domain Ω satisfying Hypothesis 2.1 yields a N . purely discrete spectrum of H0,Ω D N (iii) We recall that V is relatively form compact with respect to H0,Ω and H0,Ω , that is, D v H0,Ω − zIΩ

−1/2

N , v H0,Ω − zIΩ

−1/2

 ∈ B∞ L2 (Ω; dn x)

(4.58)

N D (in fact, much more is true as recorded , respectively, H0,Ω for all z in the resolvent sets of H0,Ω 2.31 2.32 3.47a 3.48a in (2.28) and (2.29) since B∞ can be replaced by B2p ). By (3.70) and (3.71) this yields that  the D N 2 n difference of the resolvents of HΩ and HΩ is compact (in fact, it even lies in Bp L (Ω; d x) ). By RS78 a variant of Weyl’s theorem (cf., e.g., [74, Theorem XIII.14]), oneRS78 concludes that HΩD and HΩN have D N empty essential spectrum if and only if H0,Ω and H0,Ω have (cf. [74, Problem 39, p. 369]). Thus, by part (ii) of this remark, the assumption that HΩD and HΩN have purely discrete spectra in Hypothesis h4.7 4.7 can equivalently be replaced by the assumption that Ω is bounded (still assuming Hypothesis h2.6 2.6 and that V is real-valued).

h4.7

Ya07

Assuming Hypothesis 4.7, k ∈ N, k ≥ p, we introduce (cf. also [91])     ( π −1 Im ln det k IΩ + u(H0,Ω − λIΩ )−1 v , λ ∈ (e0 , ∞)\(σ(HΩ ) ∪ σ(H0,Ω )), ξk (λ) = (4.59) 0, λ < e0 ,

4.57

where e0 = inf(σ(HΩ ), σ(H0,Ω )),

(4.60)

D and HΩ and H0,Ω temporarily abbreviate HΩD and H0,Ω in the case of Dirichlet boundary conditions N N on ∂Ω and HΩ and H0,Ω in the case of Neumann boundary conditions on ∂Ω. Moreover, we subsequently agree to write ξkD (·) and ξkN (·) for ξ(·) in the case of Dirichlet and Neumann boundary conditions in HΩ , H0,Ω . 4.57 The branch of the logarithm in (4.59) has been fixed by putting ξk (λ) = 0 for λ in a neighborhood of −∞. This is possible since   (4.61) lim det k IΩ + u(H0,Ω − λIΩ )−1 v = 1. λ↓−∞

4.58

VARIATIONS ON A THEME OF JOST AND PAIS

4.58

27

l2.3

Equation (4.61) in turn follows from Lemma 2.3 since



lim u(H0,Ω − λIΩ )−1 v

Bk (L2 (Ω;dn x))

λ↓−∞

=0

(4.62) 2.8

by applying the dominated convergence theorem to k(|·|2 − λ)−1/2 k2L2p (Rn ;dn x) as λ ↓ −∞ in (2.9) (replacing p by 2p, q by 1/2, f by u and v, etc.). Since H0,Ω is self-adjoint in L2 (Ω; dn x) with purely discrete spectrum, for any λ0 ∈ R, we obtain the norm convergent expansion −1

(H0,Ω − zIΩ )

−1

= P0,Ω,λ0 (λ0 − z)

z→λ0

+

∞ X

k+1 (−1)k S0,Ω,λ (λ0 − z)k , 0

(4.63)

4.63

k=0

where P0,Ω,λ0 denotes the Riesz projection associated with H0,Ω and the point λ0 , and S0,Ω,λ0 is given by S0,Ω,λ0 = lim (H0,Ω − zIΩ )−1 (IΩ − P0,Ω,λ0 ), (4.64) z→λ0

Ka80

with the limit taken in the topology of B(L2 (Ω; dn x)) (cf., e.g., [46, Sect. III.6]). Hence one concludes that S0,Ω,λ0 P0,Ω,λ0 = P0,Ω,λ0 S0,Ω,λ0 = 0. If, in fact, λ0 is a (necessarily discrete) eigenvalue of H0,Ω , then P0,Ω,λ0 is the projection onto the corresponding eigenspace of H0,Ω and the dimension of its range equals the multiplicity of the eigenvalue λ0 , denoted by n0,λ0 = dim(ran(P0,Ω,λ0 )).

(4.65)

We recall that all eigenvalues of H0,Ω are semisimple, that is, their geometric and algebraic multiplicities coincide, since H0,Ω is assumed to be self-adjoint. If λ0 is not in the spectrum of H0,Ω then, of course, P0,Ω,λ0 = 0 and n0,λ0 = 0. In exactly, the same manner, and in obvious notation, one then also obtains ∞ X k+1 −1 −1 (−1)k SΩ,λ (λ0 − z)k (4.66) (HΩ − zIΩ ) = PΩ,λ0 (λ0 − z) + 0 z→λ0

4.66

k=0

and nλ0 = dim(ran(PΩ,λ0 )).

(4.67)

In the following we denote half-sided limits by f (x+ ) = lim f (x + ε), ε↓0

f (x− ) = lim f (x − ε), ε↑0

x ∈ R.

(4.68)

Moreover, we denote by NHΩ (λ) (respectively, NH0,Ω (λ)), λ ∈ R, the right-continuous function on R which counts the number of eigenvalues of HΩ (respectively, H0,Ω ) less than or equal to λ, counting multiplicities. h4.7

l4.8

Lemma 4.9. Assume Hypothesis 4.7 and let k ∈ N, k ≥ p. Then ξk equals a fixed integer on any open interval in R\(σ(HΩ ) ∪ σ(H0,Ω )). Moreover, for any λ ∈ R, ξk (λ+ ) − ξk (λ− ) = −(nλ − n0,λ ),

(4.69)

4.69

and hence ξk is piecewise integer-valued on R and normalized to vanish on (−∞, e0 ) such that ξk (λ) = −[NHΩ (λ) − NH0,Ω (λ)],

λ ∈ R\(σ(HΩ ) ∪ σ(H0,Ω )).

(4.70)

4.70

Proof. Introducing the unitary operator S in L2 (Ω; dn x) of multiplication by the function sgn(V ), (Sf )(x) = sgn(V (x))f (x),

f ∈ L2 (Ω; dn x)

such that Su = uS = v, Sv = vS = u, S 2 = Isupp(V ) , one computes for λ ∈ R\σ(H0,Ω ),     det k IΩ + u(H0,Ω − λIΩ )−1 v = det k IΩ + v(H0,Ω − λIΩ )−1 u

(4.71)

4.71

28

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

  = det k IΩ + Su(H0,Ω − λIΩ )−1 vS   = det k IΩ + u(H0,Ω − λIΩ )−1 v , 

(4.72)

4.72



that is, det k IΩ + u(H0,Ω − λIΩ )−1 v is real-valued for λ ∈ R\σ(H0,Ω ). (Here the bars either denote complex conjugation, or the operator closure, depending on the context in which they are tB.3 used.) Together with the Birman–Schwinger principle as expressed in Theorem B.3, this proves that ξk equals a fixed integer on any open interval in R\(σ(HΩ ) ∪ σ(H0,Ω )). Next, we note that for z ∈ C\(σ(HΩ ) ∪ σ(H0,Ω )),   −1  d  v = tr (HΩ − zIΩ )−1 − (H0,Ω − zIΩ )−1 − ln det k IΩ + u H0,Ω − zIΩ dz (4.73)  k−1 h i`−1 X − (−1)` (H0,Ω − zIΩ )−1 v u(H0,Ω − zIΩ )−1 v u(H0,Ω − zIΩ )−1 ,

4.73

`=1

Ya07

4.63

4.66

which4.73 represents just a slight extension of the result recorded in [91]. Insertion of (4.63) and (4.66) into (4.73) then yields that for any λ0 ∈ R, ∞   X d  − ln det k IΩ + u(H0,Ω − zIΩ )−1 v = tr(PΩ,λ0 − P0,Ω,λ0 )(λ0 − z)−1 + c` (λ0 − z)` z→λ0 dz `=−k

= [nλ0 − n0,λ0 ](λ0 − z)−1 +

∞ X

c` (λ0 − z)` , (4.74)

4.74

c` ∈ R, ` ∈ Z, ` ≥ k, and c−1 = 0. (4.75) That c` ∈ R is clear from the real-valuedness of V and the self-adjointness of HΩ and H0,Ω by 4.73 expanding the (` − 1)th power of u(H0,Ω − zIΩ )−1 v in (4.73). To demonstrate P∞that c−1 actually −1 ` vanishes, that is, that the term proportional to (λ − z) cancels in the sum 0 `=−k c` (λ0 − z) in 4.74 (4.74), we temporarily introduce um = Pm u, vm = vPm , where {Pm }m∈N is a family of orthogonal projections in L2 (Ω; dn x) satisfying

4.75

z→λ0

`=−k

where

2 ∗ Pm = Pm = Pm ,

dim(ran(Pm )) = m,

ran(Pm ) ⊂ dom(v), m ∈ N,

s-lim Pm = IΩ , m↑∞

(4.76)

where s-lim denotes the limit in the strong operator topology. (E.g., it suffices to choose Pm as appropriate spectral projections associated with H0,Ω .) In addition, we introduce Vm = vm um and the operator HΩ,m in L2 (Ω; dn x) by replacing V by Vm in HΩ . Since Vm = (vPm )Pm (uPm )∗ ,

(4.77)

one obtains that Vm is a trace class (in fact, finite rank) operator, that is,  Vm ∈ B1 L2 (Ω; dn x) , m ∈ N. 2.31

(4.78)

4.77

2.32

Moreover, since by (2.28) and (2.29),  u(H0,Ω − zIΩ )−1/2 , (H0,Ω − zIΩ )−1/2 v ∈ B2p L2 (Ω; dn x) ,

z ∈ C\σ(H0,Ω ),

one concludes that Pm u(H0,Ω − zIΩ )−1 vPm = Pm u(H0,Ω − zIΩ )−1 vPm , m ∈ N, satisfies

lim Pm u(H0,Ω − zIΩ )−1 vPm − u(H0,Ω − zIΩ )−1 v B (L2 (Ω;dn x)) = 0, z ∈ C\σ(H0,Ω ), p m↑∞

lim Pm u(H0,Ω − zIΩ )−2 vPm − u(H0,Ω − zIΩ )−2 v B (L2 (Ω;dn x)) = 0, z ∈ C\σ(H0,Ω ). m↑∞

p

(4.79)

(4.80)

4.78

(4.81)

4.79

VARIATIONS ON A THEME OF JOST AND PAIS

29

Ya92

Applying the formula (cf. [90, p. 44])  d ln(detk (IH − A(z)) = −tr (IH − A(z))−1 A(z)k−1 A0 (z) , z ∈ D, (4.82) dz where A(·) is analytic in some open domain D ⊆ C with respect to the Bk (H)-norm, H a separable complex Hilbert space, one obtains for z ∈ C\(σ(HΩ ) ∪ σ(H0,Ω )),   d  − ln detk IΩ + u(H0,Ω − zIΩ )−1 v dz h i−1 h ik−1  = (−1)k tr IΩ + u(H0,Ω − zIΩ )−1 v u(H0,Ω − zIΩ )−1 v u(H0,Ω − zIΩ )−2 v , (4.83)   d  − ln detk IΩ + Pm u(H0,Ω − zIΩ )−1 vPm dz i−1 h ik−1 h Pm u(H0,Ω − zIΩ )−1 vPm = (−1)k tr IΩ + Pm u(H0,Ω − zIΩ )−1 vPm (4.84)  × Pm u(H0,Ω − zIΩ )−2 vPm , m ∈ N. 4.78

4.79

4.81

4.82

4.82

Combining equations (4.80), (4.81) and (4.83), (4.84) then yields     d  d  lim ln detk IΩ + Pm u(H0,Ω − zIΩ )−1 vPm = ln detk IΩ + u(H0,Ω − zIΩ )−1 v , m↑∞ dz dz z ∈ C\(σ(HΩ ) ∪ σ(H0,Ω )). (4.85) 4.83

4.81

4.74

Because of (4.85), to prove that c−1 = 0 in (4.74) (as claimed in (4.75)), it suffices to replace4.74 V in 4.74 (4.74) by Vm and prove that cm,−1 = 0 for all m ∈ N in the following equation analogous to (4.74),   d  − ln det k IΩ + Pm u(H0,Ω − zIΩ )−1 vPm dz ∞ (4.86) X = tr(PΩ,m,λ0 − P0,Ω,λ0 )(λ0 − z)−1 + cm,` (λ0 − z)` , m ∈ N, z→λ0

4.83

4.75

4.84

`=−k

where cm,` ∈ R, ` ∈ Z, ` ≥ k, m ∈ N,

(4.87)

4.85

and PΩ,m,λ0 denotes the corresponding Riesz projection associated with HΩ,m (obtained by replacing V by Vm in HΩ ) and the point λ0 . 4.73 Ya07 Applying the analog of formula (4.73) to HΩ,m (cf. again [91]), and noting that Pm has rank m ∈ N, one concludes that for z ∈ C\(σ(HΩ ) ∪ σ(H0,Ω )),     d  d  − ln det k IΩ + Pm u(H0,Ω − zIΩ )−1 vPm = − ln det k IΩ + Pm u(H0,Ω − zIΩ )−1 vPm dz  dz = tr (HΩ,m − zIΩ )−1 − (H0,Ω − zIΩ )−1 −

k−1 X

`

(−1) (H0,Ω −

zIΩ )−1 vPm

h

Pm u(H0,Ω −

zIΩ )−1 vPm

i`−1

−1



Pm u(H0,Ω − zIΩ )

`=1

i`  X (−1)` d h  k−1 = tr (HΩ,m − zIΩ )−1 − (H0,Ω − zIΩ )−1 − tr Pm u(H0,Ω − zIΩ )−1 vPm ` dz `=1

(4.88) −1

= tr (HΩ,m − zIΩ )

−1

− (H0,Ω − zIΩ )



4.86

30

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

+

k−1 X

 h i`−1 Pm u(H0,Ω − zIΩ )−2 vPm , (−1)` tr Pm u(H0,Ω − zIΩ )−1 vPm

m ∈ N.

`=1

4.77

Here we have used the fact that by (4.78),    d  − ln det IΩ + Pm u(H0,Ω − zIΩ )−1 vPm = tr (HΩ,m − zIΩ )−1 − (H0,Ω − zIΩ )−1 , dz Si05 for z ∈ C\(σ(HΩ ) ∪ σ(H0,Ω )), and that (cf. [82, Theorem 9.2])

(4.89)

k−1

X1 d  d d ln(detk (IH − B(z))) = ln(det(IH − B(z))) + tr B(z)` dz dz ` dz `=1

d ln(det(IH − B(z))) + = dz

k−1 X

(4.90) `−1

tr B(z)

0

 B (z) ,

z ∈ D,

`=1

where B(·) is analytic in some open domain D ⊆ C with respect to the B1 (H)-norm (with H a separable complex Hilbert space). 4.86 The presence of the d/dz-term under the sum in (4.88) proves that the only (λ0 − z)−1 -term in 4.84 4.86 (4.86), respectively, (4.88), as z → λ0 , must originate from the trace of the resolvent difference  tr (HΩ,m − zIΩ )−1 − (H0,Ω − zIΩ )−1 = tr(PΩ,m,λ0 − P0,Ω,λ0 )(λ0 − z)−1 + O(1), m ∈ N. (4.91) z→λ0

Thus we have proved that 4.84

cm,−1 = 0,

4.83

m ∈ N,

(4.92)

in (4.86). By (4.85) this finally proves c−1 = 0 (4.93) 4.74 4.74 4.75 4.69 and ( 4.75) then prove ( 4.69). Together with the paragraph following in4.72 (4.74). Equations (4.74) 4.70 (4.72), this also proves (4.70).  l4.8

t4.9

t4.2

Given Lemma 4.9, Theorem 4.3 yields the following application to differences of Dirichlet and Neumann eigenvalue counting functions:  h4.7 D ∪ Theorem 4.10. Assume Hypothesis 4.7 and let k ∈ N, k ≥ p. Then, for all λ ∈ R\ σ H Ω   D N σ H0,Ω ∪ σ H0,Ω , D (λ)] − [NH N (λ) − NH N (λ)] ξkN (λ) − ξkD (λ) = [NHΩD (λ) − NH0,Ω Ω 0,Ω      −1 ∗  −1 N − λI = π −1 Im ln detk I∂Ω − γN HΩD − λIΩ V γD H0,Ω + π −1 Im(tr(Tk (λ))) Ω  D = π −1 Im ln detk MΩD (λ)M0,Ω (λ)−1 + π −1 Im(tr(Tk (λ))) (4.94)

t4.1

with Tk defined in Theorem 4.2.

4.30

4.31

4.57

4.70

Proof. This is now an immediate consequence of (4.46), (4.47), (4.59), and (4.70). sA



Appendix A. Properties of Dirichlet and Neumann Laplacians The purpose of this appendix is to recall some basic operator domain properties of Dirichlet and h2.1 n Neumann Laplacians on sets Ω ⊂ R , n ∈ N, n ≥ 2, satisfying Hypothesis 2.1. We will show that GLMZ05 the methods developed in [31] in theh2.1 context of C 1,r -domains, 1/2 < r < 1, in fact, apply to all domains Ω permitted in Hypothesis 2.1. In this manuscript we use the following notation for the standard Sobolev Hilbert spaces (s ∈ R),   Z 2  2 s n n ∗ n b 2s H (R ) = U ∈ S(R ) | kU kH s (Rn ) = <∞ , (A.1) d ξ U (ξ) 1 + |ξ| Rn

VARIATIONS ON A THEME OF JOST AND PAIS

31

H s (Ω) = {u ∈ C0∞ (Ω)∗ | u = U |Ω for some U ∈ H s (Rn )} ,

(A.2)

H0s (Ω)

= the closure of

C0∞ (Ω)

s

in the norm of H (Ω).

(A.3)

Here C0∞ (Ω)∗ denotes the usual set of distributions on Ω ⊆ Rn , Ω open and nonempty, S(Rn )∗ is b denotes the Fourier transform of U ∈ S(Rn )∗ . It the space of tempered distributions on Rn , and U is then immediate that H s1 (Ω) ,→ H s0 (Ω) for − ∞ < s0 ≤ s1 < +∞,

(A.4)

incl-xxx

continuously and densely. Next, we recall the definition of a C 1,r -domain Ω ⊆ Rn , Ω open and nonempty, for convenience of the reader: Let N be a space of real-valued functions in Rn−1 . One calls a bounded domain Ω ⊂ Rn of class N if there exists a finite open covering {Oj }1≤j≤N of the boundary ∂Ω of Ω with the property that, for every j ∈ {1, ..., N }, Oj ∩ Ω coincides with the portion of Oj lying in the overgraph of a function ϕj ∈ N (considered in a new system of coordinates obtained from the original one via a rigid motion). Two special cases are going to play a particularly important role in the sequel. First, if N is Lip (Rn−1 ), the space of real-valued functions satisfying a (global) Lipschitz St70 condition in Rn−1 , we shall refer to Ω as being a Lipschitz domain; cf. [83, p. 189], where such domains are called “minimally smooth”. Second, corresponding to the case when N is the subspace of Lip (Rn−1 ) consisting of functions whose first-order derivatives satisfy a (global) H¨older condition of order r ∈ (0, 1), we shall say that Ω is of class C 1,r . The classical theorem of Rademacher of almost everywhere differentiability of Lipschitz functions ensures that, for any Lipschitz domain Ω, the surface measure dn−1 σ is well-defined on ∂Ω and that there exists an outward pointing normal vector ν at almost every point of ∂Ω. For a Lipschitz domain Ω ⊂ Rn it is known that ∗ H s (Ω) = H −s (Ω), −1/2 < s < 1/2. (A.5)

dual-xxx

Tr02

See [88] for this and other related properties. Next, assume that Ω ⊂ Rn is the domain lying above the graph of a function ϕ : Rn−1 → R of class 1,r C . Then for 0 ≤ s < 1+r, the Sobolev space H s (∂Ω) consists of functions f ∈ L2 (∂Ω; dn−1 σ) such that f (x0 , ϕ(x0 )), as a function of x0 ∈ Rn−1 , belongs to H s (Rn−1 ). This definition is easily adapted to the case when Ω is a domain of class C 1,r whose boundary is compact, ∗ by using a smooth partition of unity. Finally, for −1 − r < s < 0, we set H s (∂Ω)Au04 = Au06 H −s (∂Ω) . For additional background EE89 Gr85 Mc00 information in this context we refer, for instance, to [3], [4], [25, Chs. V, VI], [37, Ch. 1], [53, Ch. Wl87 3], [89, Sect. I.4.2]. To see that H 1 (∂Ω) embeds compactly into L2 (∂Ω; dn−1 σ) one can argue as follows: Given a Lipschitz domain Ω in Rn , we recall that the Sobolev space H 1 (∂Ω) is defined as the collection of functions in L2 (∂Ω; dn−1 σ) with the property that the norm of their tangential gradient belongs to L2 (∂Ω; dn−1 σ). It is essentially well-known that an equivalent characterization is that f ∈ H 1 (∂Ω) if and only if the assignment Rn−1 3 x0 7→ (ψf )(x0 , ϕ(x0 )) is in H 1 (Rn−1 ) whenever ψ ∈ C0∞ (Rn ) and ϕ : Rn−1 → R is a Lipschitz function with the propery that if Σ is an appropriate rotation and translation of {(x0 , ϕ(x0 )) ∈MM07 Rn | x0 ∈ Rn−1 }, then supp (ψ) ∩ ∂Ω ⊂ Σ. This appears to be folklore, but a proof will appear in [55, Proposition 2.4]. From the latter characterization of H 1 (∂Ω) it follows that any property of Sobolev spaces (of order 1) defined in Euclidean domains, which are invariant under multiplication by smooth, compactly supported functions as well as composition by bi-Lipschitz diffeomorphisms, readily extends to the setting of H 1 (∂Ω) (via localization and pull-back). As a concrete example, for each Lipschitz domain Ω with compact boundary, one has H 1 (∂Ω) ,→ L2 (∂Ω; dn−1 σ)

compactly.

(A.6)

EQ1

32

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

Going a bit further, we say that a domain Ω ⊂ Rn satisfies a uniform exterior ball condition (abbreviated by UEBC), if there exists R > 0 with the following property: For each x ∈ ∂Ω, there exists y = y(x) ∈ Rn such that  B(y; R) {x} ⊆ Rn \Ω and x ∈ ∂B(y; R). (A.7)

UEBC

We recall that any C 1,1 -domain (i.e., the first-order partial derivatives of the functions defining the boundary are Lipschitz) satisfies a UEBC. h2.1 e D and H eN Assuming Hypothesis 2.1, we introduce the Dirichlet and Neumann Laplacians H 0,Ω 0,Ω 2 n associated with the domain R n Ω as the unique self-adjoint operators on1 L (Ω; d x)1 whose quadratic form equals q(f, g) = Ω d x ∇f · ∇g with (form) domains given by H0 (Ω) and H (Ω), respectively. Then,   D e 0,Ω dom H = u ∈ H01 (Ω) there exists f ∈ L2 (Ω; dn x) such that (A.8) q(u, v) = (f, v)L2 (Ω;dn x) for all v ∈ H01 (Ω) ,   N 1 2 n e 0,Ω = u ∈ H (Ω) there exists f ∈ L (Ω; d x) such that dom H q(u, v) = (f, v)L2 (Ω;dn x) for all v ∈ H 1 (Ω) , (A.9) with (·, ·)L2 (Ω;dn x) denoting the scalar product in L2 (Ω; dn x). Equivalently, we introduce the densely defined closed linear operators D = ∇, dom(D) = H01 (Ω) and N = ∇, dom(N ) = H 1 (Ω) 2

n

2

n

(A.10)

n

from L (Ω; d x) to L (Ω; d x) and note that D N e 0,Ω e 0,Ω H = D∗ D and H = N ∗ N.

(A.11)

RS78

For details we refer to [74, Sects. XIII.14, XIII.15]. Moreover, with div denoting the divergence operator,  dom(D∗ ) = w ∈ L2 (Ω; dn x)n div(w) ∈ L2 (Ω; dn x) , (A.12) and hence,  D e 0,Ω dom H = {u ∈ dom(D) | Du ∈ dom(D∗ )}  (A.13) = u ∈ H01 (Ω) ∆u ∈ L2 (Ω; dn x) . One can also define the following bounded linear map ( ∗ ∗ w ∈ L2 (Ω; dn x)n div(w) ∈ H 1 (Ω) → H −1/2 (∂Ω) = H 1/2 (∂Ω) w 7→ ν · w

domHD

(A.14)

A.11

(A.15)

A.11a

by setting Z hν · w, φi =

dn x w(x) · ∇Φ(x) + hdiv(w) , Φi



whenever φ ∈ H 1/2 (∂Ω) and Φ ∈ H 1 (Ω) is such that γD Φ = φ. Here the pairing hdiv(w) , Φi A.11a ∗ in (A.15) is the natural one between functionals in H 1 (Ω) and elements in H 1 (Ω) (which, in turn, is compatible with the (bilinear) distributional pairing). It should be remarked that the above definition is independent of the particular extension Φ ∈ H 1 (Ω) of φ. Indeed, by linearity this comes down to proving that Z

hdiv(w) , Φi = − dn x w(x) · ∇Φ(x) (A.16) Ω ∗ if w ∈ L2 (Ω; dn x)n has div(w) ∈ H 1 (Ω) and Φ ∈ H 1 (Ω) has γD Φ = 0. To see this we rely on the existence of a sequence Φj ∈ C0∞ (Ω) such that Φj → Φ in H 1 (Ω). When Ω is a bounded Lipschitz j↑∞

ibp

VARIATIONS ON A THEME OF JOST AND PAIS

33

JK95

domain, this is well-known (see, e.g., [42, Remark 2.7] for a rather general result of this nature), and this result is easily extended to the case when Ω is an unbounded Lipschitz domain with a compact boundary. Indeed, if ξ ∈ C0∞ (B(0; 2)) is such that ξ = 1 on B(0; 1) and ξj (x) = ξ(x/j), j ∈ N (here B(x0 ; r0 ) denotes the ball in Rn centered at x0 ∈ Rn of radius r0 > 0), then ξj Φ → Φ in H 1 (Ω) j↑∞

and matters are reduced to approximating ξj Φ in H 1 (B(0; 2j) ∩ Ω) with test functions supported in B(0; 2j) ∩ Ω, for each fixed j ∈ N. Since γD (ξj Φ) = 0, the result for bounded Lipschitz domains applies. ibp Returning to the task of proving (A.16), it suffices to prove a similar identity with Φj in place of Φ. This, in turn, follows from of div(·) in the sense of distributions and the fact that ∗ the definition 1 1 the duality between H (Ω) and H (Ω) is compatible with the duality between distributions and test functions. Going further, one can introduce a (weak) Neumann trace operator γ eN as follows:  ∗ 1 1 −1/2 →H (∂Ω), γ eN u = ν · ∇u, (A.17) γ eN : u ∈ H (Ω) ∆u ∈ H (Ω)

A.16

A.11

with the dot product A.16 understood in the sense of (A.14). We emphasize that the weak Neumann 2.3 trace operator γ eN in (A.17) is a bounded extension of the operator γN introduced in (2.3). Indeed, to see that dom(γN ) ⊂ dom(e if u ∈ H s+1 (Ω) for some 1/2 < s < 3/2, then ∗γN ), we1 note ∗ that dual-xxx incl-xxx −1+s 1−s ∆u ∈ H (Ω) = H (Ω) ,→ H 2.3 (Ω) , by (A.5) and (A.4). With this in hand, it is then easy domHN to show that γ eN in (A.19) and γN in (2.3) agree (on the smaller domain), as claimed. We now return to the mainstream discussion. From the above preamble it follows that  dom(N ∗ ) = w ∈ L2 (Ω; dn x)n div(w) ∈ L2 (Ω; dn x) and ν · w = 0 , (A.18) A.11 eN = where the dot product operation is understood in the sense of (A.14). Consequently, with H 0,Ω ∗ N N , we have  e N = {u ∈ dom(N ) | N u ∈ dom(N ∗ )} dom H 0,Ω  = u ∈ H 1 (Ω) ∆u ∈ L2 (Ω; dn x) and γ eN u = 0 . (A.19)

domHN

D e N , where H D and H N denote the e D and H N = H =H Next, we intend to recall that H0,Ω 0,Ω 0,Ω 0,Ω 0,Ω 0,Ω 2.4 2.5 operators introduced in (2.4) and (2.5), respectively. For this purpose one can argue as follows: Mc00 D e D and Since it follows from the first Green’s formula (cf., e.g., [53, Theorem 4.4]) that H0,Ω ⊆H 0,Ω N e N . Moreover, it follows from e D and H N ⊇ H e N , it remains to show that H D ⊇ H ⊆H H0,Ω 0,Ω 0,Ω 0,Ω 0,Ω 0,Ω  2.4 domHD 2.5 domHN eD , comparing (2.4) with (A.13) and (2.5) with (A.19), that one needs only to show that dom H 0,Ω  e N ⊆ H 2 (Ω). This is the content of the next lemma. dom H 0,Ω

h2.1

lA.1

Lemma A.1. Assume Hypothesis 2.1. Then,  e D ⊂ H 2 (Ω), dom H 0,Ω

 e N ⊂ H 2 (Ω). dom H 0,Ω

(A.20)

ADNH

N N e 0,Ω H0,Ω =H .

(A.21)

DOM

Moreover, D D e 0,Ω H0,Ω =H ,

lA.1

GLMZ05

For C 1,r -domains Ω, 1/2 < r < 1, Lemma A.1 was proved in [31, Appendix A]. For bounded   Ka64 Ta65 e D ⊂ H 2 (Ω) was shown by Kadlec [44] eN ⊂ convex domains Ω, dom H and Talenti [87] and dom H 0,Ω 0,Ω GI75 H 2 (Ω) was proved by Grisvard and Ioss [38]. A unified approach to Dirichlet and Neumann problems in bounded convex domains,Mi01 which also applies to bounded Lipschitz domains satisfying UEBC, has been presented by Mitrea [57]. The extension to domains Ω with a compact boundary satisfying dual-xxx ADNH UEBC then follows as described in the paragraph following ( A.5). This establishes ( A.20) and hence DOM domHN (A.21) as discussed after (A.19).

34

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

lA.1

DHP03

We note that Lemma A.1 also followsDHP03 from [20, Theorem 8.2] in the case of C 2 -domains Ω with compact boundary. This is proved in [20] by rather different methods and can be viewed as a generalization ofGLMZ05 the classical result ADNH for bounded C 2 -domains. As shown in [31, Lemma A.2], (A.20) and (real) interpolation methods yield the following key new6.45 result (A.22) needed in the main body of this paper: h2.1

lA.2

Lemma A.2. Assume Hypothesis 2.1 and let q ∈ [0, 1]. Then for each z ∈ C\[0, ∞), one has −q −q  D N H0,Ω − zIΩ , H0,Ω − zIΩ ∈ B L2 (Ω; dn x), H 2q (Ω) . (A.22)

new6.45

Na01

Next, we recall an extension of a result of Nakamura [60, Lemma 6] from a cube A.11a in Rn to a A.16 Lipschitz domain Ω. This requires some preparation. First, we note that (A.17) and (A.15) yield the following Green formula  (A.23) he γN u, γD Φi = ∇u, ∇Φ L2 (Ω;dn x)n + h∆u, Φi, ∗ valid for any u ∈ H 1 (Ω) with ∆u ∈ H 1 (Ω) , and any Φ ∈ H 1 (Ω). The pairing on the left-hand side ∗ wGreen of (A.23) is between functionals in H 1/2 (∂Ω) and elements in H 1/2 (∂Ω), whereas the last pairing ∗ on the right-hand side is between 2.2 functionals in H 1 (Ω) and elements in H 1 (Ω). For further use, we also note that the adjoint of (2.2) maps boundedly as follows ∗ ∗ ∗ γD : H s−1/2 (∂Ω) → (H s (Ω) , 1/2 < s < 3/2. (A.24)    e N , originally defined as e N − zIΩ −1 , z ∈ C σ H Next, one observes that the operator H 0,Ω 0,Ω −1 N 2 n 2 e 0,Ω − zIΩ H : L (Ω; d x) → L (Ω; dn x), (A.25) ∗ 1 2 n can be extended to a bounded operator, mapping H (Ω) into L (Ω; d x). Specifically, since    e N ,→ H 1 (Ω) e N − zIΩ −1 : L2 (Ω; dn x) → dom H e N is bounded and since the inclusion dom H H 0,Ω 0,Ω 0,Ω  e N − zIΩ −1 as an operator is bounded, we can naturally view H 0,Ω −1 2 N b 0,Ω : L (Ω; dn x) → H 1 (Ω) (A.26) H − zIΩ mapping in a linear, bounded fashion. Consequently, for its adjoint, we have −1 ∗ ∗ N b 0,Ω H − zIΩ : H 1 (Ω) → L2 (Ω; dn x),

(A.27)

wGreen

ga*

fukcH

fukcH-bis

fukcH

and it is easy to see that this latter operator extends the one in (A.25). Hence, there is no ambiguity  fukcH-bis e N − zIΩ −1 , both for the operator in (A.27) in retaining the same symbol, that is, H as well as for 0,Ω −1 fukcH D e the operator in (A.25). Similar considerations and conventions apply to H − zIΩ . 0,Ω

lA.3

Given these preparations, we now state without proof (and for the convenience of the reader) the GLMZ05 Na01 following result proven in [31, Lemma A.3] (an extension of a result proven in [60]).    e D ∪σ H eN . Lemma A.3. Let Ω ⊂ Rn , n ≥ 2, be an open Lipschitz domain and let z ∈ C σ H 0,Ω 0,Ω Then, on L2 (Ω; dn x), −1 −1 −1 ∗ −1 D N N D e 0,Ω e 0,Ω e 0,Ω e 0,Ω H − zIΩ − H − zIΩ = H − zIΩ γD γ eN H − zIΩ , (A.28) ga*

∗ where γD is an adjoint operator to γD in the sense of (A.24)

rA.4

Remark A.4. While it is tempting to view γD as an unbounded but densely defined operator on L2 (Ω; dn x) whose domain contains the GLMZ05 space C0∞ (Ω), one should note that in this case its adjoint ∗ ∗ γD is not densely defined: Indeed (cf. [31, Remark A.4]), dom(γD ) = {0} and hence γD is not a 2 n closable linear operator in L (Ω; d x).

Na1

VARIATIONS ON A THEME OF JOST AND PAIS

h2.1

rA.5

35

lA.1

Remark A.5. In the case of domains Ω satisfying Hypothesis 2.1, Lemma A.1 implies that the e D and H e N coincide with the operators H D and H N , respectively, and hence one operators H 0,Ω 0,Ω 0,Ω 0,Ω  lA.3 D D N can use the operators H0,Ω and H0,Ω in Lemma A.3. Moreover, since dom H0,Ω ⊂ H 2 (Ω), one can 2.3 lA.3 also replace γ eN by γN (cf. (2.3)) in Lemma A.3. In particular, −1 −1  −1 ∗ −1 D N N D H0,Ω − zIΩ − H0,Ω − zIΩ = γD H0,Ω − zIΩ γN H0,Ω − zIΩ , (A.29)    D N z ∈ C σ H0,Ω ∪ σ H0,Ω , t4.1

Na1-bis

4.13

a result exploited in the proof of Theorem 4.2 (cf. (4.28)). l3.4

Finally, we prove the following result used in the proof of Lemma 3.4. lA.6

Lemma A.6. Suppose Ω ⊂ Rn , n ≥ 2, is an open Lipschitz domain with a compact, nonempty 2.2 boundary ∂Ω. Then the Dirichlet trace operator γD satisfies the following property (see also (2.2)),  γD ∈ B H (3/2)+ε (Ω), H 1 (∂Ω) , ε > 0. (A.30) Proof. First, we recall one of the equivalent definitions of H 1 (∂Ω), specifically,  H 1 (∂Ω) = f ∈ L2 (∂Ω; dn−1 σ) ∂f /∂τj,k ∈ L2 (∂Ω; dn−1 σ), j, k = 1, . . . , n ,

A.62

(A.31) A.31

where ∂/∂τk,j = νk ∂j − νj ∂k , j, k = 1, . . . , n, is a tangential derivative operator (cf. (A.33)), or equivalently,  1 2 n−1 H (∂Ω) = f ∈ L (∂Ω; d σ) there exists a constant c > 0 such that for every v ∈ C0∞ (Rn ), Z  n−1 d σf ∂v/∂τj,k ≤ c kvkL2 (∂Ω;dn−1 σ) , j, k = 1, . . . , n . (A.32)

A.64

∂Ω

Next, let u ∈ H (3/2)+ε (Ω), v ∈ C0∞ (Rn ), and ui ∈ C ∞ (Ω) ,→2.2 H (3/2)+ε (Ω), i ∈ N, be a sequence incl-xxx (3/2)+ε of functions approximating u in H (Ω). It follows from (2.2) and (A.4) that γD u, γD (∇u) ∈ L2 (∂Ω; dn−1 σ). Introducing the tangential derivative operator ∂/∂τk,j = νk ∂j −νj ∂k , j, k = 1, . . . , n, one has Z Z ∂h1 ∂h2 n−1 d σ h2 = − dn−1 σ h1 , h1 , h2 ∈ H 1/2 (∂Ω). (A.33) ∂τ ∂τ j,k j,k ∂Ω ∂Ω A.31 Utilizing (A.33), one computes for all j, k = 1, . . . , n, Z Z Z ∂v ∂ui ∂v n−1 n−1 n−1 d σ γD u = lim d σ ui = lim d σv (A.34) ∂τj,k i→∞ ∂Ω ∂τj,k i→∞ ∂Ω ∂τj,k ∂Ω Z ≤ c lim dn−1 σ v γD (∇ui ) ≤ c kγD (∇u)kL2 (∂Ω;dn−1 σ) kvkL2 (∂Ω;dn−1 σ) . i→∞

A.64

∂Ω

A.65

Thus, it follows from (A.32) and (A.34) that γD u ∈ H 1 (∂Ω). sB



Appendix B. Abstract Perturbation Theory GLMZ05

The purpose of this appendix is to summarize some of the abstract perturbation results in [31] Ka66 Ho70 KK66 which where motivated by Kato’s pioneering work [45] (see also [41], [47]) as they are needed in this paper. We introduce the following set of assumptions. hB.1

Hypothesis B.1. Let H and K be separable, complex Hilbert spaces. (i) Suppose that H0 : dom(H0 ) → H, dom(H0 ) ⊆ H is a densely defined, closed, linear operator in H with nonempty resolvent set, ρ(H0 ) 6= ∅, (B.1)

A.31

A.65

36

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

A : dom(A) → K, dom(A) ⊆ H a densely defined, closed, linear operator from H to K, and B : dom(B) → K, dom(B) ⊆ H a densely defined, closed, linear operator from H to K such that dom(A) ⊇ dom(H0 ),

dom(B) ⊇ dom(H0∗ ).

(B.2)

In the following we denote R0 (z) = (H0 − zIH )−1 , z ∈ ρ(H0 ). (B.3) (ii) Assume that for some (and hence for all ) z ∈ ρ(H0 ), the operator −AR0 (z)B ∗ , defined on dom(B ∗ ), has a bounded extension in K, denoted by K(z), K(z) = −AR0 (z)B ∗ ∈ B(K).

(B.4)

B.4

(B.5)

B.5

(iii) Suppose that 1 ∈ ρ(K(z0 )) for some z0 ∈ ρ(H0 ). (iv) Assume that K(z) ∈ B∞ (K) for all z ∈ ρ(H0 ). Ka66

Next, following Kato [45], one introduces R(z) = R0 (z) − R0 (z)B ∗ [IK − K(z)]−1 AR0 (z), tB.2

z ∈ {ζ ∈ ρ(H0 ) | 1 ∈ ρ(K(ζ))}.

hB.1

GLMZ05

Theorem B.2 ([31]). Assume Hypothesis B.1 (i)–(iii) and suppose z ∈ {ζ ∈ ρ(H0 ) | 1 ∈ ρ(K(ζ))}. B.5 Then, R(z) introduced in (B.5) defines a densely defined, closed, linear operator H in H by R(z) = (H − zIH )−1 .

(B.6)

AR(z), BR(z)∗ ∈ B(H, K)

(B.7)

B.7

R(z) = R0 (z) − R(z)B ∗ AR0 (z)

(B.8)

B.8

(z)B ∗ AR(z).

(B.9)

B.9

In addition, and

= R0 (z) − R0 ∗

Moreover, H is an extension of (H0 + B A)|dom(H0 )∩dom(B ∗ A) (the latter intersection domain may consist of {0} only), H ⊇ (H0 + B ∗ A)|dom(H0 )∩dom(B ∗ A) . (B.10) Finally, assume that H0 is self-adjoint in H. Then H is also self-adjoint if (Af, Bg)K = (Bf, Ag)K for all f, g ∈ dom(A) ∩ dom(B). tB.2

(B.11)

Ka66

In the case where H0 is self-adjoint, Theorem B.2 is due to Kato [45] in this abstract setting. The next result is an abstract version of the celebrated Birman–Schwinger principle relating eigenvalues λ0 of H and the eigenvalue 1 of K(λ0 ): hB.1

GLMZ05

tB.3

Theorem B.3 ([31]). Assume Hypothesis B.1 and let λ0 ∈ ρ(H0 ). Then, Hf = λ0 f,

0 6= f ∈ dom(H) implies K(λ0 )g = g

(B.12)

where, for fixed z0 ∈ {ζ ∈ ρ(H0 ) | 1 ∈ ρ(K(ζ))}, z0 6= λ0 , 0 6= g = [IK − K(z0 )]−1 AR0 (z0 )f = (λ0 − z0 )−1 Af.

(B.13)

Conversely, K(λ0 )g = g,

0 6= g ∈ K implies Hf = λ0 f,

(B.14)

where 0 6= f = −R0 (λ0 )B ∗ g ∈ dom(H).

(B.15)

dim(ker(H − λ0 IH )) = dim(ker(IK − K(λ0 ))) < ∞.

(B.16)

Moreover,

B.11

VARIATIONS ON A THEME OF JOST AND PAIS

37

In particular, let z ∈ ρ(H0 ), then z ∈ ρ(H) if and only if 1 ∈ ρ(K(z)). tB.3

(B.17) KK66

In the case where H0 and H are self-adjoint, Theorem B.3 is due to Konno and Kuroda [47]. Acknowledgments. We are indebted to Yuri Latushkin, Konstantin A. Makarov, and Anna Skripka for helpful discussions on this topic. We also thank Tonci Crmaric for pointing out some references to us and we are grateful to Dorina Mitrea for a critical reading of the manuscript. References ABMN05 AP04 Au04 Au06 BL06 BMN06 BC04 Bo89 BMN00

BMN02 BM04 BFK91

BFK92 BFK93 BFK95 Ca02 Ch90 Ch84 De78 DHP03 DM91 DM95 DD78 DS88 EE89

[1] S. Albeverio, J. F. Brasche, M. M. Malamud, and H. Neidhardt, Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions, J. Funct. Anal. 228, 144–188 (2005). [2] W. O. Amrein and D. B. Pearson, M operators: a generalisation of Weyl–Titchmarsh theory, J. Comp. Appl. Math. 171, 1–26 (2004). [3] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Num. Funct. Anal. Optimization 25, 321–348 (2004). [4] G. Auchmuty, Spectral charcterization of the trace spaces H s (∂Ω), SIAM J. Math. Anal. 38, 894–905 (2006). [5] J. Behrndt and M. Langer, Boundary value problems for partial differential operators on bounded domains, J. Funct. Anal. 243, 536–565 (2007). [6] J. Behrndt, M. M. Malamud, and H. Neidhardt, Scattering matrices and Weyl functions, preprint, 2006. [7] S. Blanes and F. Casas, On the convergence and optimization of the Baker–Campbell–Hausdorff formula, Lin. Algebra Appl. 378, 135–158 (2004). [8] A. Bose, Dynkin’s method of computing the terms of the Baker–Campbell–Hausdorff series, J. Math. Phys. 30, 2035–2037 (1989). [9] J. F. Brasche, M. M. Malamud, and H. Neidhardt, Weyl functions and singular continuous spectra of self-adjoint extensions, in Stochastic Processes, Physics and Geometry: New Interplays. II. A Volume in Honor of Sergio Albeverio, F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. R¨ ockner, and S. Scarlatti (eds.), Canadian Mathematical Society Conference Proceedings, Vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 75–84. [10] J. F. Brasche, M. M. Malamud, and H. Neidhardt, Weyl function and spectral properties of self-adjoint extensions, Integral Eqs. Operator Theory 43, 264–289 (2002). [11] B. M. Brown and M. Marletta, Spectral inclusion and spectral exactness for PDE’s on exterior domains, IMA J. Numer. Anal. 24, 21–43 (2004). [12] D. Burghelea, L. Friedlander, and T. Kappeler, On the determinant of elliptic differential and finite difference operators in vector bundles over S 1 , Commun. Math. Phys. 138, 1–18 (1991). Erratum: Commun. Math. Phys. 150, 431 (1992). [13] D. Burghelea, L. Friedlander, and T. Kappeler, Meyer–Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107, 34–65 (1992). [14] D. Burghelea, L. Friedlander, and T. Kappeler, Regularized determinants for pseudodifferential operators in vector bundles over S 1 , Integral Eqs. Operator Theory 16, 496–513 (1993). [15] D. Burghelea, L. Friedlander, and T. Kappeler, On the determinant of elliptic boundary value problems on a line segment, Proc. Amer. Math. Soc. 123, 3027–3038 (1995). [16] G. Carron, D´ eterminant relatif et la fonction xi, Amer. J. Math. 124, 307–352 (2002). [17] S. Chanillo, A problem in electrical prospection and a n-dimensional Borg–Levinson theorem, Proc. Amer. Math. Soc. 108, 761–767 (1990). [18] M. Cheney, Two-dimensional scattering: The number of bound states from scattering data, J. Math. Phys. 25, 1449–1455 (1984). [19] P. A. Deift, Applications of a commutation formula, Duke Math. J. 45, 267–310 (1978). [20] R. Denk, M. Hieber, and J. Pr¨ uss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166, No. 788, (2003). [21] V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95, 1–95 (1991). [22] V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73, 141–242 (1995). [23] T. Dreyfus and H. Dym, Product formulas for the eigenvalues of a class of boundary value problems, Duke Math. J. 45, 15–37 (1978). [24] N. Dunford and J. T. Schwartz, Linear Operators Part II: Spectral Theory, Interscience, New York, 1988. [25] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1989.

38

Fo87 Fr05 GL55

Ge07

GKMT01

GLMZ05 GM03

GS00 GGK00 GK69 GG91 Gr85 GI75 GG07 HL01 Ho70 JK95 JP51 Ka64 Ka66 Ka80 KK66 Ko04 KLW05 LS77 MM06 Ma04 Mc00 MPP07

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

[26] R. Forman, Functional determinants and geometry, Invent. Math. 88, 447–493 (1987). [27] L. Friedlander, Determinant of the Schr¨ odinger operator on a metric graph, preprint, 2005. [28] I. M. Gel’fand and B. M. Levitan, On the determination of a differential equation from its special function, Izv. Akad. Nauk SSR. Ser. Mat. 15, 309–360 (1951) (Russian); Engl. transl. in Amer. Math. Soc. Transl. Ser. 2, 1, 253–304 (1955). [29] F. Gesztesy, Inverse spectral theory as influenced by Barry Simon, in Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Ergodic Schr¨ odinger Operators, Singular Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory, F. Gesztesy, P. Deift, C. Galvez, P. Perry, and W. Schlag (eds.), Proceedings of Symposia in Pure Mathematics, Vol. 76/2, Amer. Math. Soc., Providence, RI, 2007, pp. 741–820. [30] F. Gesztesy, N. J. Kalton, K. A. Makarov, and E. Tsekanovskii, Some Applications of Operator-Valued Herglotz Functions, in Operator Theory, System Theory and Related Topics. The Moshe Livˇsic Anniversary Volume, D. Alpay and V. Vinnikov (eds.), Operator Theory: Advances and Applications, Vol. 123, Birkh¨ auser, Basel, 2001, p. 271–321. [31] F. Gesztesy, Y. Latushkin, M. Mitrea, and M. Zinchenko, Nonselfadjoint operators, infinite determinants, and some applications, Russ. J. Math. Phys. 12, 443–471 (2005). [32] F. Gesztesy and K. A. Makarov, (Modified ) Fredholm Determinants for Operators with Matrix-Valued SemiSeparable Integral Kernels Revisited, Integral Eqs. Operator Theory 47, 457–497 (2003). (See also Erratum 48, 425–426 (2004) and the corrected electronic only version in 48, 561–602 (2004).) [33] F. Gesztesy and B. Simon, A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure, Ann. Math. 152, 593–643 (2000). [34] I. Gohberg, S. Goldberg, and N. Krupnik, Traces and Determinants for Linear Operators, Operator Theory: Advances and Applications, Vol. 116, Birkh¨ auser, Basel, 2000. [35] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc., Providence, RI, 1969. [36] V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Kluwer, Dordrecht, 1991. [37] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985. [38] P. Grisvard, G. Iooss, Probl` emes aux limites unilat´ eraux dans des domaines non r´ eguliers, Publications des S´ eminaires de Math´ ematiques, Universit´ e de Rennes 9, 1–26 (1975). [39] C. Guillarmou and L. Guillop´ e, The determinant of the Dirichlet-to-Neumann map for surfaces with boundary, preprint, 2007. [40] P. D. Hislop and C. V. Lutzer, Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connected domains in Rd , Inverse Probl. 17, 1717–1741 (2001). [41] J. S. Howland, On the Weinstein–Aronszajn formula, Arch. Rat. Mech. Anal. 39, 323–339 (1970). [42] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130, 161–219 (1995). [43] R. Jost and A. Pais, On the scattering of a particle by a static potential, Phys. Rev. 82, 840–851 (1951). [44] J. Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain, Czechoslovak Math. J. 14, 386–393 (1964). [45] T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162, 258–279 (1966). [46] T. Kato, Perturbation Theory for Linear Operators, corr. printing of the 2nd ed., Springer, Berlin, 1980. [47] R. Konno and S. T. Kuroda, On the finiteness of perturbed eigenvalues, J. Fac. Sci., Univ. Tokyo, Sec. I, 13, 55–63 (1966). [48] S. Kotani, KdV-flow and Floquet exponent, preprint, Osaka University, 2004. [49] Y. Kurylev, M. Lassas, and R. Weder, Multidimensional Borg–Levinson theorem, Inverse Probl. 21, 1685–1696 (2005). [50] S. Levitt and U. Smilansky, A theorem on infinite products of eigenvalues of Sturm–Liouville type operators, Proc. Amer. Math. Soc. 65, 299–302 (1977). [51] M. M. Malamud and V. I. Mogilevskii, Krein type formula for canonical resolvents of dual pairs of linear relations, Methods Funct. Anal. Topology, 8, No. 4, 72–100 (2002). [52] M. Marletta, Eigenvalue problems on exterior domains and Dirichlet to Neumann maps, J. Comp. Appl. Math. 171, 367–391 (2004). [53] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [54] A. B. Mikhailova, B. S. Pavlov, and L. V. Prokhorov, Intermediate Hamiltonian via Glazman’s splitting and analytic perturbation for meromorphic matrix-functions, Math. Nachr., to appear.

VARIATIONS ON A THEME OF JOST AND PAIS

MM07 Mi96 Mi01 Mu98 NSU88 Na01 Ne72 Ne80 Ne02 Ok95 Ok95a Ot91 PS02 PW05 PW05a

Pa87 Pa02

RS75 RS79 RS78 Re03 Ry99 Sa05 Si71 Si77 Si99 Si00 Si05 St70 Su77 SU86

39

[55] I. Mitrea and M. Mitrea, Multiple Layer Potentials for Higher Order Elliptic Boundary Value Problems, preprint, 2007. [56] M. Mitrea, Boundary value problems and Hardy spaces associated to the Helmholtz equation in Lipschitz domains, J. Math. Anal. Appl. 202, 819–842 (1996). [57] M. Mitrea, Dirichlet integrals and Gaffney–Friedrichs inequalities in convex domain, Forum Math. 13, 531–567 (2001). [58] W. M¨ uller, Relative zeta functions, relative determinants and scattering theory, Commun. Math. Phys. 192, 309–347 (1998). [59] A. Nachman, John Sylvester, and G. Uhlmann, An n-dimensional Borg–Levinson theorem, Commun. Math. Phys. 115, 595–605 (1988). [60] S. Nakamura, A remark on the Dirichlet–Neumann decoupling and the integrated density of states, J. Funct. Anal. 179, 136–152 (2001). [61] R. G. Newton, Relation between the three-dimensional Fredholm determinant and the Jost function, J. Math. Phys. 13, 880–883 (1972). [62] R. G. Newton, Inverse scattering. I. One dimension, J. Math. Phys. 21, 493–505 (1980). [63] R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, New York, 2002. [64] K. Okikiolu, The Campbell–Hausdorff theorem for elliptic operators and a related trace formula, Duke Math. J. 79, 687–722 (1995). [65] K. Okikiolu, The multiplicative anomaly for determinants of elliptic operators, Duke Math. J. 79, 722–7750 (1995). [66] J. A. Oteo, The Baker–Campbell–Hausdorff formula and nested commutator identities, J. Math. Phys. 32, 419– 424 (1991). [67] L. P¨ av¨ arinta and V. Serov, An n-dimensional Borg–Levinson theorem for singular potentials, Adv. Appl. Math. 29, 509–520 (2002). [68] J. Park and K. P. Wojciechowski, Adiabatic decomposition of the ζ-determinant and Dirichlet to Neumann operator, J. Geom. Phys. 55, 241–266 (2005). [69] J. Park and K. P. Wojciechowski, Agranovich–Dynin formula for the ζ-determinants of the Neumann and Dirichlet problems, in Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, B. Booß– Bavnbek, G. Grubb, and K. P. Wojciechowski (eds.), Contemporary Math. 366, 109–121 (2005). [70] B. Pavlov, The theory of extensions and explicitly-solvable models, Russ. Math. Surv. 42:6, 127–168 (1987). [71] B. Pavlov, S-matrix and Dirichlet-to-Neumann operators, Ch. 6.1.6 in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Vol. 2, R. Pike and P. Sabatier (eds.), Academic Press, San Diego, 2002, pp. 1678–1688. [72] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [73] M. Reed and B. Simon, Methods of Modern Mathematical Physics. III: Scattering Theory, Academic Press, New York, 1979. [74] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, New York, 1978. [75] C. Remling, Inverse spectral theory for one-dimensional Schr¨ odinger operators: The A function, Math. Z. 245, 597–617 (2003). [76] V. S. Rychkov, On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc. (2) 60, 237–257 (1999). [77] M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, preprint, 2005. [78] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, NJ, 1971. [79] B. Simon, Notes on infinite determinants of Hilbert space operators, Adv. Math. 24, 244–273 (1977). [80] B. Simon, A new approach to inverse spectral theory, I. Fundamental formalism, Ann. Math. 150, 1029–1057 (1999). [81] B. Simon, Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178, 396–420 (2000). [82] B. Simon, Trace Ideals and Their Applications, Mathematical Surveys and Monographs, Vol. 120, 2nd ed., Amer. Math. Soc., Providence, RI, 2005. [83] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. [84] M. Suzuki, On the convergence of exponential operators – the Zassenhaus formula, BCH formula and systematic approximations, Commun. Math. Phys. 57, 193–200 (1977). [85] J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Commun. Pure Appl. Math. 39, 91–112 (1986).

40

SU87 Ta65 Tr02 Wl87 Ya92 Ya07

F. GESZTESY, M. MITREA, AND M. ZINCHENKO

[86] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. Math. 125, 153–169 (1987). [87] G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl. 69, 285–304 (1965). [88] H. Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers, Rev. Mat. Complut. 15, 475–524 (2002). [89] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. [90] D. R. Yafaev, Mathematical Scattering Theory, Transl. Math. Monographs, Vol. 105, Amer. Math. Soc., Providence, RI, 1992. [91] D. R. Yafaev, Perturbation determinants, the spectral shift function, trace identities, and all that, preprint, 2007. Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail address: [email protected] URL: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail address: [email protected] URL: http://www.math.missouri.edu/personnel/faculty/mitream.html Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA E-mail address: [email protected] URL: http://math.caltech.edu/∼maxim

VARIATIONS ON A THEME OF JOST AND PAIS 1 ...

VARIATIONS ON A THEME OF JOST AND PAIS. FRITZ GESZTESY, MARIUS MITREA, AND MAXIM ZINCHENKO. Abstract. We explore the extent to which a ...

455KB Sizes 0 Downloads 223 Views

Recommend Documents

Tire track geometry: variations on a theme - Department of Mathematics
theme. Serge Tabachnikov. ∗. Department of Mathematics, Penn State University. University Park ... in the ambient plane do not change; such curves are called.

outline-Reynolds-Magazine Mosaics Variations on a Theme-F17.pdf ...
outline-Reynolds-Magazine Mosaics Variations on a Theme-F17.pdf. outline-Reynolds-Magazine Mosaics Variations on a Theme-F17.pdf. Open. Extract.

Theme with Variations
The central matter of the dissertation is a context-based analysis of focus. ... definition of the theme/rheme division of sentences relative to their focus structure.

pdf-1880\variations-on-a-theme-park-the-new-american ...
Try one of the apps below to open or edit this item. pdf-1880\variations-on-a-theme-park-the-new-american-city-and-the-end-of-public-space.pdf.

Variations on a Human Face.pdf
Page 1 of 4. Name: Variations on a Human Face. Materials: 2 pennies, chart on human traits. Single Allele Traits. 1. Determine which partner will toss for the ...

Mi Pais Inventado.pdf
destruyó las torres gemelas del World Trade Center y desde ese instante algunas cosas han cambiado. No se puede. permanecer neutral en una crisis.

Comments on “Variations of tropical upper tropospheric clouds with ...
Feb 27, 2009 - RONDANELLI AND LINDZEN: CLOUD VARIATIONS WITH SST .... peratures within 0.5 C of the observed SSTs and normalizing by the sum of ...

pdf-1533\variations-on-a-shaker-melody-from-appalachian-spring ...
... of the apps below to open or edit this item. pdf-1533\variations-on-a-shaker-melody-from-appalachian-spring-from-appalachian-spring-by-unknown.pdf.

The Impact of Temperature Variations on Spectroscopic ...
The Impact of Temperature Variations on Spectroscopic Calibration Modelling: A Comparative Study. Tao Chen and Elaine Martin*. School of Chemical Engineering and Advanced Materials,. University of Newcastle, Newcastle upon Tyne, NE1 7RU, U.K.. (Submi

pdf-124\jost-nickels-groove-book-book-cd-by-jost ...
directions. Page 3 of 7. pdf-124\jost-nickels-groove-book-book-cd-by-jost-nickel.pdf. pdf-124\jost-nickels-groove-book-book-cd-by-jost-nickel.pdf. Open. Extract.

theme 1: sustainable economic growth -
Sub Theme 4: Private Sector Development, Industry and Trade . ...... efficiency in delivering postal services; and developing public online ..... increased as a share of GDP from 17.5 percent of GDP in 2006 to 22.6 percent of GDP in ..... During the

INFORMATIVO AOS PAIS Matutino.pdf
There was a problem loading more pages. Retrying... INFORMATIVO AOS PAIS Matutino.pdf. INFORMATIVO AOS PAIS Matutino.pdf. Open. Extract. Open with.

Calculus of Variations - Springer Link
Jun 27, 2012 - the associated energy functional, allowing a variational treatment of the .... groups of the type U(n1) × ··· × U(nl) × {1} for various splittings of the dimension ...... u, using the Green theorem, the subelliptic Hardy inequali

pdf-135\eighteenth-variation-from-rhapsody-on-a-theme-of-paganini ...
Try one of the apps below to open or edit this item. pdf-135\eighteenth-variation-from-rhapsody-on-a-theme-of-paganini-from-belwin-music.pdf.

Seasonal and diurnal variations of black carbon ...
Recent studies suggest that BC can alters the cloud lifetime .... Database. The first measurements using Aethalometer were ini- tially carried out on a campaign ...

[Clarinet_Institute] Beethoven, Ludwig van - 12 Variations on 'Ein ...
Whoops! There was a problem loading this page. Retrying... Page 3 of 16. KODE ETIK. Nama sumber data atau informan dalam penelitian kualitatif, tidak boleh dicantumkan apabila dapat merugikan informan tersebut. Page 3 of 16. [Clarinet_Institute] Beet

Seasonal variations of seismicity and geodetic strain in the Himalaya ...
Nov 22, 2007 - van Dam, T., Wahr, J., Milly, P.C.D., Shmakin, A.B., Blewitt, G.,. Lavallee, D., Larson, K.M., 2001. Crustal displacements due to continental water loading. Geophys. Res. Lett. 28, 651–654. Zhao, W., Nelson, K.D., project INDEPTH Tea

DownloadPDF Psychology: Themes and Variations ...
Book Synopsis. PSYCHOLOGY: THEMES AND. VARIATIONS, 10th Edition, is a fusion of the full-length and briefer versions that preceded it. The text continues ...