ON DIRICHLET-TO-NEUMANN MAPS AND SOME ...

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ON DIRICHLET-TO-NEUMANN MAPS AND SOME APPLICATIONS TO MODIFIED FREDHOLM DETERMINANTS FRITZ GESZTESY, MARIUS MITREA, AND MAXIM ZINCHENKO Dedicated with great pleasure to Boris Pavlov on the occasion of his 70th birthday

Abstract. We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schr¨ odinger operators in L2 (Ω; dn x), where Ω ⊂ Rn , n = 2, 3, are open sets with a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in L2 (Ω; dn x) to modified Fredholm perturbation determinants associated with operators in L2 (∂Ω; dn−1 σ), n = 2, 3. This leads to a two- and three-dimensional extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schr¨ odinger operator on the half-line (0, ∞) to a simple Wronski determinant of appropriate distributional solutions of the underlying Schr¨ odinger equation.

1. Introduction

s1

JP51

To describe the original Fredholm determinant result due to Jost and Pais [34], we need a few D N the one-dimensional Dirichlet and Neumann Laplacians preparations. Denoting by H0,+ and H0,+ 2 in L ((0, ∞); dx), and assuming V ∈ L1 ((0, ∞); dx), (1.1) D N we introduce the perturbed Schr¨ odinger operators H+ and H+ in L2 ((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,

(1.3)

g 0 (0) = 0, (−g 00 + V g) ∈ L2 ((0, ∞); dx)}. 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.

Date: November 11, 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 Proceedings of the conference on Operator Theory, Analysis in Mathematical Physics - OTAMP 2006, J. Janas, P. Kurasov, A. Laptev, S. Naboko, and G. Stolz (eds.), Operator Theory: Advances and Applications, Birkh¨ auser, Basel. 1

1.1

2

F. GESZTESY, M. MITREA AND M. ZINCHENKO 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 −1/2 1/2 0 φD (z, x) = z sin(z x) + dx0 z −1/2 sin(z 1/2 (x − x0 ))V (x0 )φD + + (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 dx0 z −1/2 sin(z 1/2 (x − x0 ))V (x0 )f+ (z, x0 ), f+ (z, x) = e −

(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 with (0, ∞) replaced by an appropriate open set Ω ⊂ Rn 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 (with B1 (·) abbreviating the ideal of trace class operators): 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 0 D f+ (z, 0) W (f+ (z, ·), θ+ (z, ·)) =− = . (1.12) iz 1/2 iz 1/2 1.11 JP51 Equation (1.11) is the modern formulation ofGM03 the celebrated result due to Jost and Pais [34]. N N Performing calculations similar to Section 4 in [24] for the pair of operators H0,+ and H+ , one 1.12 obtains the analogous result ( 1.12). 1.11 1.12 We emphasize that (1.11) and (1.12) exhibit a spectacular reduction of a Fredholm determinant, that is, an infinite determinant (actually, a symmetrized perturbation determinant), associated with the trace class Birmann–Schwinger kernel of a one-dimensional Schr¨odinger operator 1.4 on the half-line 1.4). This fact (0, ∞), to a simple Wronski determinant of C-valued distributional solutions of ( JP51 GM03 Ne72 Ne80 Ne02 Si00 Si05 goes back to Jost and Pais [34] (see also [24], [48], [49], [50, Sect. 12.1.2], [60], [61, Proposition 5.7], and the extensive literature cited in these references). The principal aim of this paper is to explore the extent to which this fact may generalize to higher dimensions. While a direct generalization of 1.11 1.12 (1.11), (1.12) appears to be difficult, we will next derive a formula for the ratio of such determinants which indeed permits a natural extension to higher dimensions.

1.12

DIRICHLET-TO-NEUMANN MAPS AND APPLICATIONS TO INFINITE DETERMINANTS

3

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([0, ∞)) → C, C 1 ([0, ∞)) → C, γN : (1.13) γD : g 7→ g(0), h 7→ −h0 (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 f+ (z, 0) , f+ (z, 0)

mN 0,+ (z) = − mN + (z) = −

1 = iz −1/2 , (z) mD 0,+ 1 mD + (z)

=−

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

(1.14)

1.14

(1.15)

1.15

In the case where V is real-valued, we briefly recall the spectral theoretic significance of mD + : 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 Herglotz representation represents the spectral measure of the operator H+ . In particular, dρD + (respectively,GL55 dρN determine V a.e. on (0, ∞)Re03 by the inverse spectral approachGe07 of Gelfand + ) uniquely Si99 GS00 and Levitan [20] or Simon [59], [26] (see also Remling [56] and Section 6 in the survey [21]). 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 Theorem 1.2. Assume V ∈ L1 ((0, ∞); dx) and let z ∈ C\σ(H+ ) with Im(z 1/2 ) > 0. Then, −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,+

=

t1.3

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

=

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

=

mD + (z) D m0,+ (z)

=

mN 0,+ (z) . mN + (z)

(1.17)

The proper multi-dimensional generalization to Schr¨odinger operators in L2 (Ω; dn x) corresponding to an open set Ω ⊂ Rn with compact, nonempty boundary ∂Ω then involves the operator-valued generalization of the Weyl–Titchmarsh function mD denoted by + (z), the Dirichlet-to-Neumann map 1.17 1.16 MΩD (z). Ins4particular, we will derive the following multi-dimensional extension of (1.16) and (1.17) in Section 4: h2.6 D N Theorem 1.3. Assume Hypothesis 2.6 and let z ∈ C σ HΩD ∪ σ H0,Ω ∪ σ H0,Ω . Then, −1 N − zI det 2 IΩ + u H0,Ω v Ω −1 D − zI det 2 IΩ + u H0,Ω v Ω −1 N − zI )−1 ∗ etr(T2 (z)) V γD (H0,Ω = det 2 I∂Ω − γN HΩD − zIΩ (1.18) Ω D = det 2 MΩD (z)M0,Ω (z)−1 etr(T2 (z)) . (1.19)

1.16 1.17

1.18 1.19

4

F. GESZTESY, M. MITREA AND M. ZINCHENKO

Here, det2 (·) denotes the modified Fredholm determinant in connection with Hilbert–Schmidt perturbations of the identity, T2 (z) is given by −1 ∗ −1 −1 D − zI N − zI T2 (z) = γN H0,Ω , (1.20) V HΩD − zIΩ V γD H0,Ω Ω Ω and IΩ and I∂Ω represent the identity operators in L2 (Ω; dn x) and L2 (∂Ω; dn−1 σ), respectively (with dn−1 σ the surface measure on ∂Ω). For pertinent comments on the principal reduction of (a ratio of) modified Fredholm determinants 1.18 Fredholm deterassociated with operators in L2 (Ω; dn x) on the left-hand side of (1.18) to a modified 1.18 2 n−1 1.18) and especially, minant associated with operators in L (∂Ω; d σ) on the right-hand side of ( 1.19 s4 in (1.19), we refer to Section 4. Finally, we briefly list some of the notational conventions used throughout this paper. 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 a separable complex Hilbert space H (with scalar product denoted by (·, ·)H , assumed to be linear in the second factor) 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 Bp (H), p ∈ 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 ∈ Bp (H), p ∈ N (for p = 1 we omit the subscript 1). Moreover, X1 ,→ X2 denotes the continuous embedding of the Banach space X1 into the Banach space X2 . For general references on the theory of modified Si77 FredholmSi05 determinants we refer, for instance, to DS88 GGK00 GK69 [16, Sect. XI.9], [27, Chs. IX, XI], [28, Sect. IV.2], [58], and [61, 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 , h2.1 2 n n Laplacians in L (Ω; d x) associated with open sets Ω ⊂ R , n = 2, 3, introduced in Hypothesis 2.1 −q D,N below. In particular, we study mapping properties of H0,Ω − zIΩ , q ∈ [0, 1], (IΩ the identity −q D,N operator in L2 (Ω; dn x)) and trace ideal properties of the maps f H0,Ω − zIΩ , f ∈ Lp (Ω; dn x), −s −r N D for appropriate p ≥ 2, and γN H0,Ω − zIΩ , and γD H0,Ω − zIΩ , for appropriate r > 3/4, s2.2 > 1/4, with γ and γD being the Neumann and Dirichlet boundary trace operators defined in 2.3 N (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 = 2, 3 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. h2.1

The class of domains described in Hypothesis 2.1 is a subclass of all Lipschitz domains with compact nonempty boundary. We also note that while ∂Ω is assumed to be compact, Ω may be unbounded (e.g., an exterior domain) in connection with conditions (i) or (iii). For more details in

DIRICHLET-TO-NEUMANN MAPS AND APPLICATIONS TO INFINITE DETERMINANTS

5

thisGLMZ05 context, in particular, for the precise definition of the uniform exterior ball condition, we refer GMZ06 Gr85 GI75 JK81 Ka64 to [23, App. A] and [25, App. A] (and the references cited therein, such as [30, Ch. 1], [31], [32], [35], Mc00 Mi96 Mi01 St70 Ta65 Tr02 Wl87 [40, Ch. 3], [43], [44], [62, p. 189], [63], [68], and [70, Sect. I.4.2]). 0 First, we introduce the boundary trace operator γD (Dirichlet trace) by 0 γD : C(Ω) → C(∂Ω),

0 γD u = u|∂Ω .

(2.1)

Mc00

Then there exists a bounded, linear operator γD (cf. [40, 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 (∂Ω) ,→ L2 (∂Ω; dn−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 their domains. We recall that dn−1 σ denotes the surface measure on ∂Ω andTr02 we GLMZ05 Mc00 referWl87 to [23, App. A] for our notation in connection with Sobolev spaces (see also [40, Ch. 3], [68], and [70, Sect. I.4.2]). 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 N D associated with the domain Ω as follows, and H0,Ω Laplacians H0,Ω D D H0,Ω = −∆, dom H0,Ω = {u ∈ H 2 (Ω) | γD u = 0}, (2.4) N N 2 H0,Ω = −∆, dom H0,Ω = {u ∈ H (Ω) | γN u = 0}. (2.5) GLMZ05

2.4 2.5

RS78

D N A detailed discussion of H0,Ω and H0,Ω is provided in [23, App. A] (cf. also [55, Sects. X.III.14, X.III.15]).

h2.1

l2.2

2.4

N D introduced in (2.4) and and H0,Ω Lemma 2.2. Assume Hypothesis 2.1. Then the operators H0,Ω 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 byGLMZ05 the spectral theorem for self-adjoint operators. JK95 Mc00 MMT01 MT00 Ta91 [23, Lemma A.2] (based on results in [33], [40, Thm. 4.4,Ve84 App. B], [42], [45], [64, As explained in Ta96 Tr78 Chs. 1, 2], [65, Props. 4.5, 7.9], [66, Sect. 1.3, Thm. 1.18.10, Rem. 4.3.1.2], [69]), the key ingredients l2.2 in proving Lemma 2.2 are the inclusions D N dom H0,Ω ⊂ H 2 (Ω), dom H0,Ω ⊂ H 2 (Ω) (2.7) and real interpolation methods. GLMZ05 The next result is a slight extension of [23, Lemma 6.8] and provides an explicit discussionGMZ06 of the GLMZ05 z-dependence of the constant c appearing in estimate (6.48) of [23]. For a proof we refer to [25]. GMZ06

l2.3

h2.1

Lemma 2.3 ([25]). 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)

2.7

6

F. GESZTESY, M. MITREA AND M. ZINCHENKO

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 2 −q 2 ≤c 1+ 2q k(| · | − z) kLp (Rn ;dn x) kf kLp (Ω;dn x) , 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

(2.9)

2.8

(Here, in obvious notation, (| · |2 − z)−q denotes the function (|x|2 − z)−q , x ∈ Rn .) Next we recall certain boundedness 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 − 1+ε D N 4 4 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

(2.10)

2.22

2.3

As in [23, 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 f1 ∈ Bp1 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , (2.11) γD H0,Ω Ω −1 D − zI γN H0,Ω f2 ∈ Bp2 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) (2.12) 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

(2.13)

2.27

(2.14)

2.28

σ))

l2.3

l2.4

As in [23, 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 4/3 < p ≤ 2, in the case n = 2, and 3/2 < p ≤ 2, in the case n = 3. h2.6

Assuming Hypothesis 2.6, we next introduce the perturbed operators HΩD and HΩN in L2 (Ω; dn x) Ka66 KK66 by alluding to abstract perturbation results due to Kato [36] (see also Konno and Kuroda [37]) as GLMZ05 summarized in [23, Sect. 2]: Let V , u, and v denote the operators of multiplication by functions V , u = exp(i arg(V ))|V |1/2 , and v = |V |1/2 in L2 (Ω; dn x), respectively, such that V = uv. 2p

n

(2.15)

2.29

z ∈ C\[0, ∞),

(2.16)

2.31

z ∈ C\[0, ∞),

(2.17)

2.32

l2.3

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) , Ω −1/2 −1/2 N N − zI u H0,Ω − zIΩ , H0,Ω v ∈ B2p L2 (Ω; dn x) , Ω

DIRICHLET-TO-NEUMANN MAPS AND APPLICATIONS TO INFINITE DETERMINANTS

7

and hence, in particular, N 1/2 N dom(u) = dom(v) ⊇ dom H0,Ω = H 1 (Ω) ⊃ H 2 (Ω) ⊃ dom H0,Ω , D 1/2 D dom(u) = dom(v) ⊇ H 1 (Ω) ⊃ H01 (Ω) = dom H0,Ω ⊃ dom H0,Ω . 2.31

(2.18) (2.19)

2.32

Moreover, (2.16) and (2.17) imply −1 −1 D − zI N − zI u H0,Ω v, u H0,Ω v ∈ Bp L2 (Ω; dn x) ⊂ B2 L2 (Ω; dn x) , Ω Ω

z ∈ C\[0, ∞). (2.20) 2.8 l2.3 Utilizing ( 2.9) in Lemma 2.3 with −z > 0 sufficiently large, such that the B -norms of the operators 2p 2.31 2.32 in (2.16) and (2.17) are less than 1, one concludes that the Hilbert–Schmidt norms of the operators 2.35 GLMZ05 in (2.20) are less than 1. Thus, applying [23, Thm. 2.3], one obtains the densely defined, closed D D N operators HΩD and HΩN (which are extensions of H0,Ω +V defined on dom H0,Ω ∩dom(V ) and H0,Ω + N D V defined on dom H0,Ω ∩ dom(V ), respectively). In particular, the resolvent of HΩ (respectively, HΩN ) is explicitly given by −1 i−1 −1 −1 −1 −1 h D D D D − zI v u H0,Ω − zIΩ , HΩD − zIΩ = H0,Ω − zIΩ − H0,Ω − zIΩ v IΩ + u H0,Ω Ω z ∈ C\σ HΩD , (2.21) i−1 h −1 −1 −1 −1 −1 N N N N − zI v u H0,Ω − zIΩ , HΩN − zIΩ = H0,Ω − zIΩ − H0,Ω − zIΩ v IΩ + u H0,Ω Ω z ∈ C\σ HΩN . (2.22) i h −1 D,N − zIΩ v for z ∈ ρ HΩD,N is guaranteed by arguments disHere invertibility of IΩ + u H0,Ω GLMZ05 cussed, for instance, in [23, Sect. 2] and the literature cited therein. Although we will not explicitly use the following result in this paper, we feel it is of sufficient independent interest to be included at the end of this section:

2.35

2.20

2.21

h2.1

L-1

Lemma 2.7. Assume Ω satisfies Hypothesis 2.1 with n = 2, 3 replaced by n ∈ N, n ≥ 2, suppose that V ∈ Ln/2 (Ω; dn x), and let  (0, 1) if n = 2,       [0, 3 ) if n = 3, 2 s∈ (2.23)  [0, 2) if n = 4,      [0, 2] if n ≥ 5.

SSS

Then the operator ∗ V : H s (Ω) → H 2−s (Ω)

(2.24)

V-2

(2.25)

V-3

is well-defined and bounded, in fact, it is compact. V-2

Proof. The fact that the operator (2.24) is well-defined along with the estimate kV kB(H s (Ω),(H 2−s (Ω))∗ ) ≤ C(n, Ω)kV kLn/2 (Ω) , Tr83

are direct consequences of standard embedding results (cf. [67, Sect. 3.3.1] for smooth domains and Tr02 V-2 [68] for arbitrary (bounded or unbounded) Lipschitz domains). Once the boundedness of (2.24) has been established, the compactness follows from the fact that if Vj ∈ C0∞ (Ω) is a sequence of functions ∗ V-3 with the property that Vj → V in Ln/2 (Ω), then Vj → V in B H s (Ω), H 2−s (Ω) by (2.25) j↑∞ j↑∞ ∗ ET89 and each operator Vj : H s (Ω) → H 2−s (Ω) is compact, by Rellich’s selection lemma (cf. [17] for

8

F. GESZTESY, M. MITREA AND M. ZINCHENKO

Tr02

V-2

smooth domains and [68] for arbitrary Lipschitz domains). Thus, the operator in (2.24) is compact as the operator norm limit of a sequence of compact operators.

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 ∗ γ eN : u ∈ H 1 (Ω) ∆u ∈ H 1 (Ω) → H −1/2 (∂Ω) (3.1) 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

Ω

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 particular extension Φ of φ, and that γ eGLMZ05 N is an extension of the Neumann trace operator γN 2.3 defined in (2.3). For more details we refer to [23, App. A]. We start with a basic result on the Helmholtz Dirichlet and Neumann boundary value problems: h2.1 GMZ06 D Theorem 3.1 ([25]). 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 N γD H0,Ω − zIΩ : L2 (∂Ω; dn−1 σ) → H 3/2 (Ω), z ∈ C σ H0,Ω . (3.8)

Finally, the solutions

uD 0

and

uN 0

3.1

3.3a

3.2

3.4a

3.4b 3.4c

are given by the formulas

−1 ∗ D uD f, 0 (z) = − γN H0,Ω − zIΩ −1 ∗ N N u0 (z) = γD H0,Ω − zIΩ g. t3.1

GMZ06

A detailed proof of Theorem 3.1 will appear in [25]. We temporarily strengthen our hypothesis on V and introduce the following assumption:

(3.9)

3.3

(3.10)

3.4

DIRICHLET-TO-NEUMANN MAPS AND APPLICATIONS TO INFINITE DETERMINANTS

9

h2.1

h3.2

Hypothesis 3.2. Suppose the set Ω satisfies Hypothesis 2.1 and assume that V ∈ L2 (Ω; dn x) ∩ Lp (Ω; dn x) for some p > 2.

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), (3.11) γD uD = f on ∂Ω,

t3.1

D has a unique eN uD ∈ L2 (∂Ω; dn−1 σ). Similarly, for every g ∈ L2 (∂Ω; dn−1 σ) solution u satisfying γ N and z ∈ C σ HΩ the following Neumann boundary value problem, ( (−∆ + V − z)uN = 0 on Ω, uN ∈ H 3/2 (Ω), (3.12) γ eN uN = g on ∂Ω,

has a unique solution uN . Moreover, the solutions uD and uN are given by the formulas −1 ∗ ∗ uD (z) = − γN HΩD − zIΩ f, (3.13) −1 ∗ ∗ uN (z) = γD HΩN − zIΩ g. (3.14) D ∪ σ HΩD in the case of the Dirichlet problem Proof. We temporarily assume that z ∈ C σ H0,Ω N N and z ∈ C σ H0,Ω ∪ σ HΩ in the context of the Neumann problem. Uniqueness follows from the fact that z ∈ / σ(HΩD ) and z ∈ / σ(HΩN ), respectively. Next, we will show that the functions −1 D D uD (z) = uD V u0 (z), (3.15) 0 (z) − HΩ − zIΩ −1 N N N N u (z) = u0 (z) − HΩ − zIΩ V u0 (z), (3.16) t3.1

3.11

3.10

3.11 3.12

3.13 3.14

3.12

N with uD 0 , ut3.1 0 given by Theorem 3.1, satisfy (3.13) and (3.14), respectively. Indeed, it follows from N 3/2 2 n−1 Theorem 3.1 that uD (Ω) and γ eN uD σ). Using the Sobolev embedding 0 , u0 ∈ H 0 ∈ L (∂Ω; d 3/2 q n theorem H (Ω) ,→ L (Ω; d x), q ≥ 2, and3.13 the fact that V ∈ Lp (Ω; dn x), p > 2, one concludes 3.14 D N 2 n that V u0 , V u0 ∈ L (Ω; d x), and hence (3.15) and (3.16) are well-defined. Since one also has −1 −1 l2.3 D N V ∈ L2 (Ω; dn x), it follows from Lemma 2.3 that V H0,Ω −zIΩ and V H0,Ω −zIΩ are Hilbert– Schmidt, and hence −1 −1 D D I + V H0,Ω − zIΩ ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω ∪ σ HΩD , (3.17) −1 −1 N N I + V H0,Ω − zIΩ ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω ∪ σ HΩN . (3.18)

2.4

3.9

2.5

Thus, by (2.4) and (2.5), −1 D −1 −1 −1 D D D HΩD − zIΩ V u0 = H0,Ω − zIΩ I + V H0,Ω − zIΩ V u0 ∈ H 2 (Ω), −1 N −1 −1 −1 N N N HΩN − zIΩ V u0 = H0,Ω − zIΩ I + V H0,Ω − zIΩ V u0 ∈ H 2 (Ω),

(3.19) (3.20)

and hence uD , uN ∈ H 3/2 (Ω) and γ eN uD ∈ L2 (∂Ω; dn−1 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

(3.21) −1

V uN 0

3.15 3.16

10

F. GESZTESY, M. MITREA AND M. ZINCHENKO N = V uN 0 − IΩ V u0 = 0,

2.4

2.5

3.15

(3.22)

3.16

and by (2.4), (2.5) and (3.17), (3.18) one also obtains, −1 D D γD uD = γD uD V u0 0 − γD HΩ − zIΩ −1 −1 D −1 D D = f − γD H0,Ω − zIΩ I + 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 I + V H0,Ω − zIΩ V u0 = g. 3.11

3.12

3.3

3.4

3.13

(3.23)

(3.24)

3.14

Finally, (3.13) and (3.14) follow from (3.9), (3.10), (3.15), (3.16), 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 f IΩ − HΩD − zIΩ V −1 ∗ ∗ D = − γN HΩ − zIΩ f, (3.25) −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. (3.26) Analytic z ∈ / continuation with respect to z then permits one to remove the additional condition N D in the in the case of the Dirichlet problem, and the additional condition z ∈ / σ H0,Ω σ H0,Ω context of the Neumann problem. h3.2

D Assuming Hypothesis 3.2, we now introduce the Dirichlet-to-Neumann maps, M0,Ω (z) and MΩD (z), associated with (−∆ − z) and (−∆ + V − z) on Ω, as follows, ( H 1 (∂Ω) → L2 (∂Ω; dn−1 σ), D D M0,Ω (z) : z ∈ C σ H0,Ω , (3.27) D f 7→ −e γN u0 ,

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,

(3.28)

z ∈ C σ HΩD ,

(3.29)

and MΩD (z) :

( H 1 (∂Ω) → L2 (∂Ω; dn−1 σ), f 7→ −e γN uD ,

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

uD ∈ H 3/2 (Ω),

γD uD = f on ∂Ω.

(3.30)

h3.2

N In addition, still assuming Hypothesis 3.2, we introduce the Neumann-to-Dirichlet maps, M0,Ω (z) N and MΩ (z), associated with (−∆ − z) and (−∆ + V − z) on Ω, as follows, ( L2 (∂Ω; dn−1 σ) → H 1 (∂Ω), N N M0,Ω (z) : z ∈ C σ H0,Ω , (3.31) 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,

(3.32)

DIRICHLET-TO-NEUMANN MAPS AND APPLICATIONS TO INFINITE DETERMINANTS

11

and MΩN (z) :

( L2 (∂Ω; dn−1 σ) → H 1 (∂Ω), g 7→ γD uN ,

z ∈ C σ HΩN ,

(3.33)

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.

(3.34)

h3.2

It follows from Theorems 3.1 and 3.3, that under the assumption of Hypothesis 3.2, the operators D N M0,Ω (z), MΩD (z), M0,Ω (z), and MΩN (z) are well-defined and satisfy the following equalities, N D D N M0,Ω (z) = −M0,Ω (z)−1 , z ∈ C σ H0,Ω ∪ σ H0,Ω , (3.35) N D −1 D N MΩ (z) = −MΩ (z) , z ∈ C σ HΩ ∪ σ HΩ , (3.36)

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 ∗ ∗ N N N MΩ (z) = γD γD HΩ − zIΩ , z ∈ C σ HΩ . 3.30

(3.37)

3.30

(3.38)

3.31

(3.39)

3.32

(3.40)

3.33

3.33

The representations (3.37)–(3.40) provide a convenient point of departure for proving the operatorwill 3.33 return to this topic in a future paper. valued Herglotz property of MΩD and MΩN . We 3.30 Next, we note that the above formulas (3.37)–(3.40) may be used as alternative definitions of 3.31 the Dirichlet-to-Neumann and Neumann-to-Dirichlet maps. In particular, we will next use ( 3.38) 3.33 and (3.40) to extend the above definition of the operators MΩD (z) and MΩN (z) to the more general h2.6 situation governed by Hypothesis 2.6: h2.6

GMZ06

l3.4

Lemma 3.4 ([25]). Assume Hypothesis 2.6. Then the operators MΩD (z) and MΩN (z) defined by 3.31 3.33 equalities (3.38) and (3.40) have the following boundedness properties, MΩD (z) ∈ B H 1 (∂Ω), L2 (∂Ω; dn−1 σ) , z ∈ C σ HΩD , (3.41) N 2 n−1 1 N MΩ (z) ∈ B L (∂Ω; d σ), H (∂Ω) , z ∈ C σ HΩ . (3.42) l3.4

3.42a 3.43a

GMZ06

A detailed proof of Lemma 3.4 will be provided in [25]. 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, relevant in the context ABMN05 BL06BMN06 BMN00 BMN02 BM04 of boundary value spaces (boundary triples, etc.), we refer, for instance, to [1], [3], [4], [5], [6], [7], DM91 DM95 GKMT01GG91 MM06 Ma04 MPP07 Pa87 Pa02 [14], [15], [22], [29, Ch. 3], [38], [39], [41], [52], [53]. t4.2 Next, we prove the following auxiliary result, which will play a crucial role in Theorem 4.2, the principal result of this paper. h2.6

l3.5

Lemma 3.5. 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 , −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 .

(3.43)

3.35

(3.44)

3.36

12

F. GESZTESY, M. MITREA AND M. ZINCHENKO

Proof. Let z ∈ C identity

3.35 3.30 3.31 D σ H0,Ω ∪ σ HΩD . Then (3.43) follows from (3.37), (3.38), and the resolvent

−1 −1 ∗ ∗ D H0,Ω − zIΩ − HΩD − zIΩ −1 −1 ∗ ∗ D − zI HΩD − zIΩ V H0,Ω (3.45) 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 (3.35), (3.39), and (3.43) 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 (3.46) Ω −1 ∗ ∗ N × γD γD H0,Ω − zIΩ . D M0,Ω (z) − MΩD (z) = γ eN γN eN γN =γ

3.40

t3.1

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

(3.47)

3.41

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

u ∈ H 3/2 (Ω),

γ eN u = g on ∂Ω.

(3.48)

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 (3.47) and (3.49) that −1 ∗ ∗ −1 ∗ ∗ −1 ∗ ∗ D N N γN H0,Ω − zIΩ γD γD H0,Ω − zIΩ = − γD H0,Ω − zIΩ . 3.44

(3.49)

3.43

(3.50)

3.44

3.43

3.40

3.36

Finally, insertion of (3.50) into (3.46) yields (3.44). 4.24

4.29

3.35

eN can be replaced by γN on the right-hand side of (3.43) and It follows from (4.25)–(4.30) that γ 3.36 (3.44). 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)

In particular, H1 and H2 may have different dimensions. Especially, one of them may be infinite and 4.0 the other finite, in which case one of the two determinants in (4.1) reduces to a finite determinant. JP51 [34] as described This case indeed occurs in the original one-dimensional case studied by Jost and Pais GM03 in detail in [24] and the references therein. In the proof of the next theorem, the role of H1 and H2 will be played by L2 (Ω; dn x) and L2 (∂Ω; dn−1 σ), respectively. GLMZ05 We start with an extension of a result in [23]:

4.0

DIRICHLET-TO-NEUMANN MAPS AND APPLICATIONS TO INFINITE DETERMINANTS

h2.6

t4.1

Theorem 4.1. Assume Hypothesis 2.6 and let z ∈ C D − zI γN H0,Ω Ω

−1

D N σ HΩD ∪ σ H0,Ω ∪ σ H0,Ω . Then,

−1 −1 ∗ N − zI V γD H0,Ω ∈ B1 L2 (∂Ω; dn−1 σ) , Ω −1 −1 ∗ N − zI HΩD − zIΩ V γD H0,Ω ∈ B2 L2 (∂Ω; dn−1 σ) , Ω

V HΩD − zIΩ

γN

13

(4.2)

4.1

(4.3)

4.2

(4.4)

4.3

and −1 N − zI v det 2 IΩ + u H0,Ω Ω −1 D − zI det 2 IΩ + u H0,Ω v Ω −1 −1 ∗ N − zI = det 2 I∂Ω − γN HΩD − zIΩ V γD H0,Ω Ω −1 −1 −1 ∗ N − zI D − zI V γD H0,Ω V HΩD − zIΩ × exp tr γN H0,Ω . Ω Ω 4.3

2.35

Proof. From the outset we noteNthat the left-hand side of (4.4) is well-defined by (2.20). Let D z ∈ C σ HΩD ∪ σ H0,Ω ∪ σ H0,Ω and 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)|

(4.5) ,

(4.6)

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

n = 2, n = 3,

( 3p, p2 = 4p,

n = 2, n=3

(4.7)

h2.6

with p as introduced in Hypothesis 2.6. Then it follows that p11 + p12 = p1 , in both cases n = 2, 3, and hence V = uv = u eve. Next, we introduce −1 −1 D − zI N − zI v, KN (z) = −u H0,Ω v (4.8) KD (z) = −u H0,Ω Ω Ω and note that [IΩ − KD (z)]−1 ∈ B L2 (Ω; dn x) ,

z∈C

D σ HΩD ∪ σ H0,Ω .

(4.9)

Thus, utilizing the following facts, [IΩ − KD (z)]−1 = IΩ + KD (z)[IΩ − KD (z)]−1

(4.10)

1 = det 2 [IΩ − KD (z)][IΩ − KD (z)]−1 = det 2 IΩ − KD (z) det 2 [IΩ − KD (z)]−1 exp tr KD (z)2 [IΩ − KD (z)]−1 ,

(4.11)

and

one obtains det 2 [IΩ − KN (z)][IΩ − KD (z)]−1 = det 2 IΩ − KN (z) det 2 [IΩ − KD (z)]−1 exp tr KN (z)KD (z)[IΩ − KD (z)]−1 det 2 IΩ − KN (z) exp tr (KN (z) − KD (z))KD (z)[IΩ − KD (z)]−1 . = det 2 IΩ − KD (z)

(4.12)

4.6

14

F. GESZTESY, M. MITREA AND M. ZINCHENKO

4.3

At this point, the left-hand side of (4.4) can be rewritten as −1 N − zI det 2 IΩ + u H0,Ω v Ω det 2 IΩ − KN (z) −1 = det I − K (z) D − zI 2 Ω D det 2 IΩ + u H0,Ω v Ω = det 2 [IΩ − KN (z)][IΩ − KD (z)]−1 × exp tr (KD (z) − KN (z))KD (z)[IΩ − KD (z)]−1 = det 2 IΩ + (KD (z) − KN (z))[IΩ − KD (z)]−1

−1

× exp tr (KD (z) − KN (z))KD (z)[IΩ − KD (z)]

(4.13)

4.12

.

GLMZ05

Next, temporarily suppose that V ∈ Lp (Ω; dnGLMZ05 x) ∩ L∞ (Ω; dn x). Using [23, Lemma A.3] (an extension Na01 of a result of Nakamura [47, Lemma 6]) and [23, Remark A.5], one finds D −1 −1 N − zI KD (z) − KN (z) = −u H0,Ω − zIΩ − H0,Ω v Ω −1 −1 ∗ N − zI D − zI (4.14) γN H0,Ω v = −u γD H0,Ω Ω Ω h i∗ −1 −1 N − zI D − zI = − γD H0,Ω u γN H0,Ω v. Ω Ω 4.13

4.13

4.12

Thus, inserting (4.14) into (4.13) yields, −1 N − zI det 2 IΩ + u H0,Ω v Ω −1 D − zI det 2 IΩ + u H0,Ω v Ω h −1 i∗ −1 h −1 i−1 N − zI D − zI D − zI = det 2 IΩ − γD H0,Ω u γ H v I + u H v Ω N Ω Ω Ω 0,Ω 0,Ω h i −1 −1 ∗ N − zI D − zI × exp tr γD H0,Ω v u γN H0,Ω Ω Ω h −1 −1 i−1 D − zI D − zI × u H0,Ω v IΩ + u H0,Ω v . (4.15) Ω Ω c2.5

4.6

Then, utilizing Corollary 2.5 with p1 and p2 as in (4.7), 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

(4.16) (4.17)

and hence, h −1 i∗ −1 N − zI D − zI γD H0,Ω u γN H0,Ω v ∈ Bp L2 (Ω; dn x) ⊂ B2 L2 (Ω; dn x) , Ω Ω −1 h −1 i∗ D − zI N − zI γN H0,Ω u ∈ Bp L2 (∂Ω; dn−1 σ) ⊂ B2 L2 (∂Ω; dn−1 σ) . v γD H0,Ω Ω Ω

(4.19)

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

(4.20)

4.0

z∈C

D σ HΩD ∪ σ H0,Ω , 4.14

(4.18)

one now applies the idea expressed in formula (4.1) and rearranges the terms in (4.15) as follows: −1 N − zI det 2 IΩ + u H0,Ω v Ω −1 D − zI det 2 IΩ + u H0,Ω v Ω

4.14

DIRICHLET-TO-NEUMANN MAPS AND APPLICATIONS TO INFINITE DETERMINANTS

15

−1 h −1 i−1 h −1 i∗ D − zI D − zI N − zI = det 2 I∂Ω − γN H0,Ω v IΩ + u H0,Ω v γD H0,Ω u Ω Ω Ω −1 −1 D − zI D − zI × exp tr γN H0,Ω v v u H0,Ω Ω Ω h i h −1 −1 −1 i∗ D − zI N − zI × IΩ + u H0,Ω v γD H0,Ω u Ω Ω i h h −1 −1 −1 i∗ −1 D − zI D − zI N − zI e H0,Ω ve γD H0,Ω u e = det 2 I∂Ω − γN H0,Ω ve IΩ + u Ω Ω Ω −1 −1 D − zI D − zI × exp tr γN H0,Ω e H0,Ω ve u ve (4.21) Ω Ω h −1 i−1 h −1 i∗ D − zI N − zI × IΩ + u e H0,Ω ve γD H0,Ω u e . Ω Ω

4.20

In the last equality we employed the following simple identities, V = uv = u eve, h h −1 i−1 −1 i−1 D − zI D − zI v IΩ + u H0,Ω v u = ve I + u e H0,Ω ve u e. Ω Ω

(4.22) (4.23)

4.20

Utilizing (4.21) and the following resolvent identity, HΩD − zIΩ

−1

D − zI ve = H0,Ω Ω

−1 h −1 i−1 D − zI ve IΩ + u e H0,Ω ve , Ω

(4.24)

4.23

4.3

one arrives at (4.4), subject to the extra assumption V ∈GLMZ05 Lp (Ω; dn x) ∩ L∞ (Ω; dn x). l2.3 p n Finally, assuming only V ∈ L (Ω; d x) and utilizing [23, Thm. 3.2], Lemma 2.3, and Corollary c2.5 2.5 once again, one obtains h −1 i−1 D − zI IΩ + u e H0,Ω ve ∈ B L2 (Ω; dn x) , (4.25) Ω −p/p1 D u e H0,Ω − zIΩ ∈ Bp1 L2 (Ω; dn x) , (4.26) −p/p 2 D ve H0,Ω − zIΩ ∈ Bp2 L2 (Ω; dn x) , (4.27) −1 N − zI γD H0,Ω u e ∈ Bp1 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , (4.28) Ω −1 D − zI γN H0,Ω ve ∈ Bp2 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , (4.29) Ω

4.24 4.25 4.26 4.27 4.28

and hence 4.23

D − zI u e H0,Ω Ω

4.29

4.1

−1

ve ∈ Bp L2 (Ω; dn x) ⊂ B2 L2 (Ω; dn x) . 4.2

(4.30) 4.3

Relations (4.24)–(4.30) prove (4.2) and (4.3), and 2.8 hence,2.27 the leftand the right-hand sides of (4.4) 2.28 are well-defined for V ∈ Lp (Ω; dn x). Thus, using (2.9), (2.13), (2.14), the continuity of det 2 (·) with , the continuity of tr(·) with respect to the respect to the Hilbert–Schmidt norm k · k B2 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)

Vk ∈ Lp (Ω; dn x) ∩ L∞ (Ω; dn x), k ∈ N, in the norm of Lp (Ω; dn x) as k ↑ ∞, then extends the result from V ∈ Lp (Ω; dn x) ∩ L∞ (Ω; dn x) to V ∈ Lp (Ω; dn x), n = 2, 3.

t4.2

Given these preparations, wet1.2 are now ready for the principal result of this paper, the multidimensional analog of Theorem 1.2: h2.6 D N Theorem 4.2. Assume Hypothesis 2.6 and let z ∈ C σ HΩD ∪ σ H0,Ω ∪ σ H0,Ω . Then, D MΩD (z)M0,Ω (z)−1 − I∂Ω = −γN HΩD − zIΩ

−1 N − zI )−1 ∗ ∈ B L2 (∂Ω; dn−1 σ) (4.31) V γD (H0,Ω Ω 2

4.29

16

F. GESZTESY, M. MITREA AND M. ZINCHENKO

and −1 N − zI det 2 IΩ + u H0,Ω v Ω −1 D − zI det 2 IΩ + u H0,Ω v Ω −1 N − zI )−1 ∗ etr(T2 (z)) = det 2 I∂Ω − γN HΩD − zIΩ V γD (H0,Ω Ω D D −1 tr(T2 (z)) = det 2 MΩ (z)M0,Ω (z) e ,

(4.32)

4.30

(4.33)

4.31

where D − zI T2 (z) = γN H0,Ω Ω

−1

V HΩD − zIΩ

−1 ∗ −1 N − zI ∈ B1 L2 (∂Ω; dn−1 σ) . (4.34) V γD H0,Ω Ω l3.5

t4.1

Proof. The result follows from combining Lemma 3.5 and Theorem 4.1.

A few comments are in order at this point. 4.30 4.31 The sudden appearance of the exponential term exp(tr(T2 (z))) in (4.32) and (4.33), when compared to the one-dimensional case, is due to the necessary use of the modified determinant detp (·) t4.1 t4.2 in Theorems 4.1 and 4.2. 4.30 1.16 The multi-dimensional extension (4.32) of (1.16), under the stronger hypothesis V ∈ L2 (Ω; dn x), GLMZ05 t4.2 n = 2, 3,GLMZ05 first appeared in [23]. However, the present results in Theorem 4.2h2.1 go decidedly beyond those in [23] in the sense that the class of domains Ω permitted by Hypothesis 2.1 is greatly enlarged GLMZ05 h2.6 as compared to [23] and the conditions on V satisfying Hypothesis 2.6 are nearly optimal bySi71 comparCh84 RS75 and Simon [54, Sect. IX.4], Simon [57, Sect. ison with the Sobolev inequality (cf. Cheney [13], Reed 4.31 1.17 I.1]). Moreover, the multi-dimensional extension (4.33) of (1.17) invoking Dirichlet-to-Neumann maps is a new result. t4.2 The principal reduction in Theorem 4.2 reduces (a ratio of) 4.30 modified Fredholm determinants associated with operators in L2 (Ω; dn x) on the left-hand side of (4.32) to modified Fredholm deter4.30 2 n−1 minants associated with operators in L (∂Ω; d σ) on the right-hand side of ( 4.32) and especially, 4.31 in (4.33). 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 has Fo87 been studied by Forman [18]. 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 ant1.1 analog of the one-dimensional Jost and Pais formula mentioned in the introduction (cf. Theorems 1.1 and t1.2 1.2). Since then, this topic has been further developed in various directions and we refer, for instance, BFK91 BFK92 BFK93BFK95 Ca02 Fr05 Mu98 to Burghelea, Friedlander,PW05 and Kappeler [8], [9], [10], [11], Carron [12], Friedlander [19], M¨ uller [46], Park and Wojciechowski [51], and the references therein. 4.30

c4.3

Remark 4.3. The following observation yields a simple application of formula (4.32). Since by GLMZ05 the Birman–Schwinger principle (cf., e.g., the discussion in [23, Sect. 3]), for any z ∈ C σ HΩD ∪ −1 D N N − zI σ H0,Ω ∪ σ H0,Ω , one has z ∈ σ HΩN if and only if det 2 IΩ + u H0,Ω v = 0, it follows Ω 4.30 from (4.32) that D N for all z ∈ C σ HΩD ∪ σ H0,Ω ∪ σ H0,Ω , one has z ∈ σ HΩN (4.35) −1 −1 ∗ N − zI if and only if det 2 I∂Ω − γN HΩD − zIΩ V γD H0,Ω = 0. Ω

DIRICHLET-TO-NEUMANN MAPS AND APPLICATIONS TO INFINITE DETERMINANTS

17

4.30

One can also prove the following analog of (4.32): −1 D − zI det 2 IΩ + u H0,Ω v Ω −1 N − zI det 2 IΩ + u H0,Ω v Ω −1 −1 ∗ D − zI V γD (HΩN − zIΩ )∗ = det 2 I∂Ω + γN H0,Ω Ω −1 −1 −1 ∗ D − zI N − zI × exp − tr γN H0,Ω V HΩN − zIΩ V γD H0,Ω . Ω Ω Then, proceeding as before, one obtains 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 2 I∂Ω + γN H0,Ω V γD HΩN − zIΩ = 0. Ω

(4.36)

(4.37)

Acknowledgments. We are indebted to Yuri Latushkin and Konstantin A. Makarov for numerous discussions on this topic. We also thank the referee for a careful reading of our manuscript and his constructive comments. Fritz Gesztesy would like to thank all organizers of the international conference on Operator Theory and Mathematical Physics (OTAMP), and especially, Pavel Kurasov, for their kind invitation, the stimulating atmosphere during the meeting, and the hospitality extended to him during his stay in Lund in June of 2006. He also gratefully acknowledges a research leave for the academic year 2005/06 granted by the Research Council and the Office of Research of the University of Missouri–Columbia. References ABMN05 AP04 BL06 BMN06 BMN00

BMN02 BM04 BFK91

BFK92 BFK93 BFK95 Ca02 Ch84

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