Quadratic eigenvalue problems for second order systems



Sonja Currie † < [email protected] > Bruce A. Watson ‡ < [email protected] > School of Mathematics University of the Witwatersrand Private Bag 3, P O WITS 2050, South Africa May 17, 2011

Abstract We consider the spectral structure of a quadratic second order system boundaryvalue problem. In particular we show that all but a finite number of the eigenvalues are real and semi-simple. We develop the eigencurve theory for these problems and show that the order of contact between an eigencurve and the parabola gives the Jordan chain associated with the eigenvector corresponding to that eigencurve. Following this we use our knowledge of the eigencurves to obtain eigenvalue asymptotics. Finally the completeness of the eigenfunctions is studied using operator matrix techniques. It should be noted here that the usual left definiteness assumptions have been overcome in this study. ∗ Keywords: Quadratic pencils, eigencurves, completeness, eigenvalue asymptotics (2000)MSC: 34B07, 34L10, 34L20, 34B45 . † Supported by NRF Thutuka grant no. TTK2007040500005 ‡ Supported in part by the Centre for Applicable Analysis and Number Theory. Supported by NRF grant no. FA2007041200006

1

1

Introduction

There is a considerable literature on the Sturm-Liouville quadratic pencil on a finite interval. The majority of this literature requires left-definiteness (see section 2) of the associated Sturm-Liouville operator. This assumption ensures that the eigenvalues are all real and algebraically simple. For abstract versions of this problem see [5] and [27, 28] where completeness and eigenfunction expansions are considered. Similar types of problems were also studied in [2, 23, 25, 29] and in [14, 16, 18] where eigenvalue asymptotics were given. In [6] a Sturm oscillation theorem and eigenvalue asymptotics were given without the left-definiteness restriction. Here we study a class of quadratic problems for second order systems, distinct from those considered previously in that we do not require the left-definiteness assumption noted earlier. In the final section we apply these results to quadratic Sturm-Liouville pencils on finite graphs with general self-adjoint boundary conditions at the nodes. In section 2, we give the system eigenvalue problem that we are considering in this paper together with some preliminaries. The nature of the spectrum is considered. In particular we show that the number of non-real eigenvalues (counted according to multiplicity) is finite and we provide a bound on the non-real eigenvalues. The eigencurve theory is the focus of section 3. In this setting the eigenvalues are the intersections between the eigencurves µn (λ), n = 0, 1, . . . , and the parabola µ = λ2 . We relate Jordan chain lengths to order of contact of the eigencurve µn (λ) with the parabola µ = λ2 at eigenvalues. The eigencurve theory leads to eigenvalue asymptotics for the quadratic system boundary value problem in section 4. In section 5, we show that the eigenvectors of the quadratic system problem are almost doubly complete in L2 [0, 1]. To do this we reformulate the quadratic operator eigenvalue problem as a self-adjoint linear operator matrix problem in a Pontryagin space. We thank the referees for their valuable advice.

2

Preliminaries

The focus of this paper is the quadratic eigenvalue problem −(M (x)Y 0 )0 + Q(x)Y − λS(x)Y = λ2 Y,

(2.1)

where Q(x) and M (x) are n × n hermitian matrices for each x and each component of Q(x) and M −1 (x) is an L1 (0, 1) function. Here M (x) is positive definite, and S(x) is an hermitian matrix valued function with entries from L∞ (0, 1). We impose boundary conditions of the form A∗ Y (0) − B ∗ M Y 0 (0) = 0, ∗



0

Γ Y (1) − ∆ M Y (1) = 0,

2

(2.2) (2.3)

where A, B, Γ, ∆ are n × n-matrices with A∗ A + B ∗ B = I, Γ∗ Γ + ∆∗ ∆ = I, A∗ B = B ∗ A and Γ∗ ∆ = ∆∗ Γ. With the above boundary conditions the differential expression, −(M (x)Y 0 )0 +Q(x)Y , is formally self-adjoint, and has a self-adjoint operator realizaton, T . Let T be the matrix Sturm-Liouville operator given by T Y = −(M Y 0 )0 + QY, with domain D(T ) = {Y | Y, M Y 0 ∈ AC[0, 1], T Y ∈ L2 [0, 1], Y obeys (2.2), (2.3)}. The problem (2.1)-(2.3) is said to be left-definite if T is a positive definite operator. Let h , i be the inner product given by hF, Gi =

n Z X i=1

1

Z

1

Fi Gi dt =

F T G dt.

(2.4)

0

0

By [1] and [31], T is self-adjoint (with respect to the above inner product), has compact resolvent and the spectrum of T consists only of point spectrum, is real, countably infinite and has +∞ as its only accumulation point. Let (τj ) denote the eigenvalues of T indexed so that τ0 ≤ τ1 ≤ τ2 ≤ . . . , i.e. arranged in increasing order and repeated according to geometric multiplicity. Note that, as the operator T is self-adjoint, its eigenvalues are real and semi-simple, in other words the algebraic and geometric multiplicities coincide, or equivalently all Jordan chains are of length 1. Let S : L2 [0, 1] → L2 [0, 1] be the bounded multiplication operator given by (SY )(x) = S(x)Y (x). Let s¯ and s respectively denote the supremum and infimum of the spectrum of S. We recall from [24] the following characterization of the algebraic multiplicity of an eigenvalue specialized to the case of the quadratic pencil. W (λ) := T − λS − λ2 I. Definition 2.1 (i) An eigenvalue of (2.1)-(2.3) is a complex number λ0 for which (2.1)-(2.3) has a non-trivial solution and such a non-trivial solution is called an eigenvector to the eigenvalue λ0 . The number of linearly independent eigenvectors to the eigenvalue λ0 is called the geometric multiplicity of the eigenvalue λ0 .

3

(ii) An eigenvalue λ0 with eigenvector ϕ0 6= 0 of (2.1)-(2.3) is said to support a Jordan chain of length m ≥ 1 if there are functions ϕ1 , . . . , ϕm−1 ∈ D(T ) with   i X 1 ∂j W (λ) ϕi−j = 0, j! ∂λj λ=λ0

i = 1, . . . , m − 1,

(2.5)

j=0

but no function ϕm ∈ D(T ) can be found to satisfy (2.5) with i = m. The maximal value of the sum of the lengths of the Jordan chains supported by each linearly independent family of eigenvectors at λ0 is the algebraic multiplicty of the eigenvalue λ0 . (iii) The eigenvalue λ0 is said to be semi-simple if its algebraic and geometric multiplicities coincide, i.e. if each Jordan chain has at most length 1. Theorem 2.2 The boundary value problem (2.1)-(2.3) has countably infinitely many eigenvalues, they have no finite accumulation point and each eigenvalue has finite algebraic multiplicity. In addition all but finitely many eigenvalues of (2.1)-(2.3) are real. √ If λ is an eigenvalue which is non-real then |λ| ≤ −τ0 . Proof: Consider the equation −(M Y 0 )0 + Q(x)Y − λδS(x)Y = λ2 Y,

δ ∈ [0, 1],

(2.6)

with boundary conditions (2.2), (2.3). If δ = 0 the above boundary value problem reduces to the eigenvalue problem T Y = ρY , where ρ = λ2 and we know that T has compact resolvent. The ρ-spectrum is point spectrum, real, countably infinite and has only finitely many negative eigenvalues. As only finitely many negative ρ-eigenvalues occur only finitely many non-real λ-eigenvalues may appear. The only accumulation point of the ρ-spectrum is +∞ giving that the λ-spectrum accumulates at ±∞. Let Y (λ, x) be a fundamental solution of (2.6) with Y (0) = B and Y 0 (0) = A, then Y (λ, x) obeys the boundary condition (2.2) and is an entire function in δ. Let Dδ denote the characteristic determinant Dδ = det(∆∗ M Y 0 (1) − Γ∗ Y (1)) then Dδ has zeros located at the eigenvalues of (2.1)-(2.3) with order of the zero equal to the algebraic multiplicity of the eigenvalue, see [24]. Let λ0 be an eigenvalue of (2.1)-(2.3), by contradiction we show that the algebraic multiplicity of λ0 is finite. Assume that the algebraic multiplicity of λ0 is infinite. Then D1 (λ) ≡ 0 since the order of the zero at λ = λ0 of D1 (λ) is infinite and D1 vanishes everywhere. Thus every λ ∈ C is an eigenvalue. In particular λ = iθ, θ ∈ R, is eigenvalue and we can solve (non-trivially) T Y − iθSY = (iθ)2 Y

4

with solution say Yθ . Thus T Yθ + θ2 Yθ = iθSYθ , and < T Yθ + θ2 Yθ , Yθ >=< iθSYθ , Yθ >= i < θSYθ , Yθ > . Here, for θ large, the left hand side is positive and the right hand side is pure imaginary, thus we have a contradiction. Hence the algebraic multiplicity of an eigenvalue of (2.1)(2.3) is finite. Also since D1 6≡ 0 we have that (2.1)-(2.3) has at most countably many eigenvalues. By [17], each eigenvalue λn (0) of the problem, repeated by multiplicity, is continuous in δ. So there are infinitely many eigenvalues for any given δ, i.e. Dδ has infinitely many zeros for any given δ. Since the zeros of Dδ have no finite accumulation point, as Dδ 6≡ 0, there are countably many eigenvalues and only finitely many of the zeros of Dδ for any √ given δ can occur in |λ| ≤ −τ0 . Hence there exists infinitely many eigenvalues λn (1) √ outside of |λ| ≤ −τ0 . Now proceeding as in the proof of [6, Theorem 3.1] we get that √ all the eigenvalues outside of |λ| ≤ −τ0 are real.

3

Eigencurves

Let λ0 ∈ R. For λ ∈ R we can define real-analytic functions µn (λ) and Zn (λ), n = 0, 1, . . ., where µn (λ) is an eigenvalue and Zn (λ) a corresponding eigenfunction of the operator −T + λS, see [17, page 392]. Here (Zn (λ0 )) is an orthonormal sequence and µn (λ0 ), n = 0, 1, . . . are an increasing sequence with repetition according to geometric multiplicity. Then Zn (λ) ∈ D(T ) and setting Wn (λ) = T − Sλ + µn (λ)I, 2

W (λ) = T − λS − λ I,

(3.1) (3.2)

we have Wn (λ)Zn (λ) = 0.

(3.3)

The curves µn (λ) will be referred to as the eigencurves of the two-parameter problem (T − λS + µI)Y = 0,

0 6= Y ∈ D(T ),

(3.4)

associated with (2.1)-(2.3). The eigenvalues of (2.1)-(2.3) can now be given as the λvalues at which some eigencurve µ = µn (λ), n = 0, 1, . . . , and the parabola µ = −λ2 intersect. Definition 3.1 Two functions F and G, real analytic in a neighbourhood of λ0 are said to have order of contact k at λ0 if F (i) (λ0 ) = G(i) (λ0 ), i = 0, . . . , k, and F (k+1) (λ0 ) 6= G(k+1) (λ0 ).

5

(j)

Lemma 3.2 For Zn as defined above, Zn (λ0 ) ∈ D(T ), j = 0, 1, . . . , and (T Zn )(j) (λ0 ) = (j) (j) T Zn (λ0 ) where Zn is the j th λ derivative of Zn , and T Zn(j) (λ)

 j  X j (Sλ − µn (λ))(j−i) Zn(i) (λ). = i i=0

Proof: By the definition of Zn it follows that the lemma holds for j = 0. Suppose the result holds for j. For λ 6= λ0 we have (j)

(j)

(j)

(j)

Zn (λ) − Zn (λ0 ) ∈ D(T ). λ − λ0 As Z is real analytic so lim

λ→λ0

Zn (λ) − Zn (λ0 ) = Zn(j+1) (λ0 ). λ − λ0

Also ! (j) (j) Zn (λ) − Zn (λ0 ) lim T λ→λ0 λ − λ0  j  (i) (i) X (Sλ − µn (λ)I)(j−i) Zn (λ) − [(Sλ − µn (λ)I)(j−i) Zn ]λ=λ0 j = lim i λ→λ0 λ − λ0 i=0 j  X

(i)

(i)

(Sλ − µn (λ)I)(j−i) Zn (λ) − (Sλ − µn (λ)I)(j−i) Zn (λ0 ) = lim λ→λ0 λ − λ0 i=0   j (i) (i) X (Sλ − µn (λ)I)(j−i) Zn (λ0 ) − [(Sλ − µn I)(j−i) ]λ=λ0 Zn (λ0 ) j + lim i λ→λ0 λ − λ0 i=0  j  i X j h = (Sλ − µn (λ)I)(j−i) Zn(i+1) (λ) + (Sλ − µn (λ)I)(1+j−i) Zn(i) (λ) i λ=λ0 =

i=0 j+1 X i=0

j+1 i

j i



h

i (Sλ − µn (λ)I)(1+j−i) Zn(i) (λ) (j)

As T is closed it now follows that (j)

j+1 

X ∂Zn T (λ) = ∂λ i=0

∂Zn ∂λ

λ=λ0

∈ D(T ) and

j+1 i



(Sλ − µn (λ)I)(1+j−i) Zn(i) (λ).

Thus the result holds for j + 1, and by induction, in general. Lemma 3.3 Let j = k, . . . , m be the j values for which µj (λ0 ) = −λ20 . If eigencurves µj (λ), j = k, . . . , m, and the parabola µ = −λ2 have intersections with order of contact

6

νj , where νk ≤ νk+1 ≤ . . . ≤ νm , at the eigenvalue λ0 of (2.1)-(2.3), then the Jordan chain structure of the eigenspace at λ0 consists of m − k + 1 Jordan chains with lengths mk , . . . , mm where mj ≥ νj + 1, j = k, . . . , m. Proof: By definition of the eigencurves, Zk (λ0 ), . . . , Zm (λ0 ) span the geometric eigenspace of the eigenvalue λ0 . For n ∈ {k, . . . , m}, we now consider the Jordan chain based at [r] (r) the eigenvector Zn (λ0 ). Setting Yn = r!1 Zn (λ0 ), and observing that h i h i ∂rW (r) 2 (r) = − (λS − µ (λ)) (λ ) = − (λS + λ ) n 0 ∂λr λ=λ0 λ=λ0 for 1 ≤ r ≤ νn + 1 we have, upon taking successive derivatives via the previous lemma, that the eigencurve equation (3.3) yields  j  X j W (i) (λ0 )Zn(j−i) (λ). 0= i i=0

Thus for r = 0, . . . , νj + 1, 0=

j X 1 (i) W (λ0 )Yn[j−i] (λ), i! i=0

[0]

[νn ]

and Yn , . . . , Yn

[0]

is a Jordan chain for λ0 based at Yn .

Theorem 3.4 Let j = k, . . . , m be the j values for which µj (λ0 ) = −λ20 . If eigencurves µj (λ), j = k, . . . , m, and the parabola µ = −λ2 have intersections with order of contact νj , where νk ≤ νk+1 ≤ . . . ≤ νm , at the eigenvalue λ0 of (2.1)-(2.3), then a maximal linearly independent Jordan chain structure of the eigenspace at λ0 consists of m − k + 1 Jordan chains with lengths mk ≤ . . . ≤ mm where mj = νj + 1, j = k, . . . , m. Proof: Let Y [0] , . . . , Y [`] , be a Jordan chain of length ` + 1 associated with the eigenvalue λ0 . Suppose that ` ≥ νi +1, then as the order of contact between µ = µi (λ) and µ = −λ2 is νi at λ0 we have that (νi +1)

W (νi +1) (λ0 ) − Wi

(λ0 ) = βi I

where βi ∈ R\{0}. Also (j)

W (j) (λ0 ) = Wi (λ0 ),

for j = 0, . . . , νi .

(3.5)

We focus our attention on the subchain Y [0] , . . . , Y [νi +1] . Here (νi +1)

βi (Y [0] , Zi (λ0 )) = (W (νi +1) Y [0] , Zi (λ0 )) − (Y [0] , Wi

7

Zi (λ0 ))

(3.6)

Now as Y [0] , . . . , Y [νi +1] is a Jordan chain and as Wi Zi (λ) = 0 for λ in a neighbourhood of λ0 we have νX i +1 W (j) (λ0 ) [νi +1−j] Y =0 j! j=0

and

νX i +1 j=0

(ν +1−j)

(j)

Wi (λ0 ) Zi i =0 j! (νi + 1 − j)! (j)

which when applied along with the self-adjointness of W (j) (λ0 ) and Wi (λ0 ) to (5.5) give !# ! " νi (j) (νi +1−j) (j) (λ ) X W Z (λ ) −βi W 0 0 i (Y [0] , Zi (λ0 )) = Y [νi +1−j] , Zi (λ0 ) − Y [0] , i (νi + 1)! j! j!(νi + 1 − j)! j=0 ! !# " νi (j) [0] Z (νi +1−j) (λ ) (j) Z (λ ) X W (λ )Y W 0 0 i 0 i − = , i Y [νi +1−j] , j! j! (νi + 1 − j)! j=0 ! !# " νi (j) (ν +1−j) X (λ0 ) W (j) (λ0 )Y [0] Zi i [νi +1−j] Wi Zi (λ0 ) = − , Y , , j! j! (νi + 1 − j)! j=0

by (3.5). As W (λ0 )Y [0] = 0 and Wi Zi (λ0 ) = 0 the term in the summation resulting from j = 0 is zero, giving " ! !# νi (j) (ν +1−j) X (λ0 ) −βi W (j) (λ0 )Y [0] Zi i [νi +1−j] Wi Zi (λ0 ) [0] Y , (Y , Zi (λ0 )) = , − (νi + 1)! j! j! (νi + 1 − j)! j=1

Now using

r X W (j) (λ0 )

j!

j=0

and

Y [r−j] = 0

r [r−j] (j) X Wi (λ0 ) Zi =0 j! (r − j)! j=0

for r = 0, . . . , νi , gives j−1 νi (r) (j−r) X X Wi Zi (λ0 ) βi (Y [0] , Zi (λ0 )) = Y [νi +1−j] , (νi + 1)! r!(j − r)! r=0

j=1



=

νi X

j−1 X

j=1

r=0

νX i −1

!

νi X

r=0 j=r+1

8

(ν +1−j)

(λ0 ) W (r) (λ0 )Y [j−r] Zi i , r! (νi + 1 − j)! ! (r) (j−r) (λ0 ) [νi +1−j] Wi Zi Y , r!(j − r)!

!



νX i −1

νi X

r=0 j=r+1

(ν +1−j)

W (r) Zi i (λ0 ) Y [j−r] , r!(νi + 1 − j)!

! .

In the former of these sums we replace j by s = νi + 1 − j and in the latter we replacing j by s = j − r, to obtain ! νX (r) (νi +1−s−r) i −1 ν i −r X W Z (λ ) βi 0 i (Y [0] , Zi (λ0 )) = Y [s] , i (νi + 1)! r!(νi + 1 − s − r)! r=0 s=1 ! νX i −1 ν i −r (r) Z (νi +1−r−s) (λ ) X 0 [s] W i − Y , r!(νi + 1 − r − s)! r=0 s=1

= 0. As βi 6= 0, we have (Y [0] , Zi (λ0 )) = 0 for all i with νi < `. Let X = span(Zk (λ0 ), . . . , Zm (λ0 )) which has dimension d = 1+m−k. List the values of νi , i = 1, . . . , N by n1 < n2 < . . . < nN where N ≤ d. Set Λj = {i|νi = nj }, j = 1, . . . , N . [0] [m −1] Let Yi , . . . , Yi i , i = k, . . . , m, be a maximal linearly independent Jordan chain structure for the eigenspace at λ0 ordered so that m1 ≤ m2 ≤ . . . ≤ md . From Lemma 3.3, mi ≥ nj + 1 = νi + 1 for i ∈ Λj , it is thus adequate to prove mi ≤ nj + 1, i ∈ Λj . [0]

If mi > nN + 1 then Yi ⊥ Zj (λ0 ) for all j = k, . . . , m, which is not possible as Y [0] ∈ X\{0}. Thus mi ≤ nN + 1 for all i = k, . . . , m, making mi = nN + 1 = νi + 1 for i ∈ ΛN . [0]

[0]

−1 N −1 Λr and mi > nN −1 + 1 then Yi ⊥ Zj (λ0 ) for all j ∈ ∪N If i ∈ ∪r=1 r=1 Λr . Hence Yi [0] [0] is a linear combination of Zr (λ0 ), r ∈ ΛN , giving Yi and Yj , j ∈ ΛN as #(ΛN ) + 1 linearly independent elements in the space spanned by Zr (λ0 ), r ∈ ΛN , a contradiction. Thus mi ≤ nN −1 + 1 for all i ∈ {k, . . . , m}\ΛN . Hence mi = νi + 1 for all i ∈ ΛN −1 .

Assume that mi = νi + 1 for all i ∈ Λj , j = J, . . . , N. If i ∈ ∪J−1 r=1 Λr and mi > nJ−1 + 1 [0] [0] J−1 then Yi ⊥ Zj (λ0 ) for all j ∈ ∪r=1 Λr . Hence Yi is a linear combination of Zr (λ0 ), r ∈ P [0] [0] ∪N and Yj , j ∈ ∪N r = J N #(Λr ) linearly independent t=J Λt , giving Yi r=J Λr as 1 + elements in the space spanned by Zr (λ0 ), r ∈ ∪N r=J Λr , a contradiction. Thus mi ≤ nJ−1 + 1 and mr = νr for all r ∈ ΛJ−1 . Thus proving the proposition by induction. It can be shown (using the above theorem) that if Y [0] , . . . , Y [`] is a Jordan chain at λ0 , then these chain elements can be represented as Y

[j]

=

j m X X i=k r=0

(j−r)

ari

Zi , (j − r)!

where ari = 0 for all ` − νi > r.

9

j = 0, . . . , `,

4

Asymptotics

In this section we begin by giving bounds on the slope of the eigencurves, from these we deduce that all large eigenvalues are real and semi-simple. Finally we give asymptotic estimates for the eigenvalues of the boundary value problem given by (2.1)-(2.3). Lemma 4.1 If µ = µk (λ) is an eigencurve of (2.1)-(2.3) for the real eigenvalue λk then dµk (λ) s≤ ≤ s. (4.1) dλ λ=λ0 Proof: To reduce the number of subscripts in the proof we denote µk (λ) by µ and the corresponding real-analytic family of eigenfunctions Zk (λ) normalized at λk by Z. By definition −(M Z 0 )0 + QZ − λSZ = −µZ. (4.2) On taking the derivative of (4.2) with respect to λ we obtain −(M Zλ0 )0 + QZλ − SZ − λSZλ = −µZλ − µλ Z.

(4.3)

Using the approach of Richardson in [26], we multiply the hermitian adjoint of (4.2) by Zλ on the right hand side and (4.3) by Z ∗ on the left hand side to get −(Z ∗ 0 M )0 Zλ + Z ∗ QZλ − λZ ∗ SZλ = −µZ ∗ Zλ ,

(4.4)

−Z ∗ (M Zλ0 )0 + Z ∗ QZλ − Z ∗ SZ − λZ ∗ SZλ = −µZ ∗ Zλ − µλ Z ∗ Z.

(4.5)

So subtracting (4.4) from (4.5) gives −Z ∗ (M Zλ0 )0 + (Z ∗ 0 M )0 Zλ − Z ∗ SZ = −µλ Z ∗ Z.

(4.6)

Integrating (4.6) by parts from 0 to 1 with respect to x gives Z 1 Z 1  1 (Z ∗ SZ − µλ Z ∗ Z) dx = (M Z 0 )∗ Zλ − Z ∗ (M Zλ0 ) 0 − Z ∗ 0 M Zλ0 − Z ∗ 0 M Zλ0 dx 0 0   0 ∗ ∗ 0 1 = (M Z ) Zλ − Z (M Zλ ) 0 . (4.7) Let c(λ) = B ∗ Z(0) + A∗ M Z 0 (0), then c(λ), λ ∈ R, is a real-analytic function of λ with Z(0) = Bc(λ)

and M Z 0 (0) = Ac(λ),

so Zλ (0) = Bcλ (λ)

and M Zλ0 (0) = Acλ (λ).

Similary setting b(λ) = ∆∗ Z(1)+Γ∗ M Z 0 (1), then b(λ), λ ∈ R, is a real-analytic function of λ with Z(1) = ∆b(λ) and M Z 0 (1) = Γb(λ),

10

so Zλ (1) = ∆bλ (λ)

and M Zλ0 (1) = Γbλ (λ).

Now as B ∗ A = A∗ B and ∆∗ Γ = Γ∗ ∆, it follows from the above that the right hand side of (4.7) is 0. Thus Z 1 Z 1 ∗ Z ∗ SZ dx, µλ Z Z dx = 0

0

giving R1

Z ∗ SZ dx hZ, SZi hZ, SZi µλ = R0 1 = = . ∗ hZ, Zi ||Z||2 0 Z Z dx Here s¯ ≥

hZ, SZi ≥ s. ||Z||2

Hence s ≤ µλ ≤ s. Theorem 4.2 If λ ∈ R is an eigenvalue of (2.1)-(2.3) and λ > eigenvalue λ is semi-simple.

−s 2

or λ <

−s 2

then the

Proof: Suppose that the eigenvalue λ is non-semisimple and lies on eigencurves µk , . . . , µm . From Theorem 3.4, the order of contact between µ = −λ2 and at least one of these eigencurves, say µm , is at least 1. Thus −2λ = µm λ . Now from Lemma 4.1 s ≤ µm λ ≤ s giving s ≤ −2λ ≤ s from which the theorem follows. Theorem 4.3 The large modulus eigenvalues of the boundary value problem given by (2.1)-(2.3) when repeated according multiplicity take on the asymptotic form √ λ± n → ∞, (4.8) n = ± τn + O(1), where τn , n ∈ N, are as in section 2. Proof: From Lemma 4.1, the eigencurve µk (λ) which has µk (0) = −τk must lie between the straight lines given by µ√= sλ−τk and µ√ = sλ−τk . These lines intersect the parabola µ = −λ2 at the points lie in

−s±

s2 +4τk 2

−s±

s2 +4τ

k and . Thus for λ > 0 the intersection will 2 " # p p −s + s2 + 4τk −s + s2 + 4τk I1 := , 2 2

and for λ < 0 in

" I2 :=

−s −

# p p s2 + 4τk −s − s2 + 4τk , . 2 2

For large λ any contact between the eigencurve µk and the parabola −λ2 is simple, and an intersections take place once in each of the intervals I1 and I2 . Hence for large k the two eigenvalues on the eigencurve µk are √ λ± k = ± τk + O(1), where the order estimate is as k → ∞.

11

5

Completeness

As T is a self-adjoint operator with compact resolvent in L2 [0, 1], the eigenvectors form an orthonormal basis for L2 [0, 1]. In particular if ei , i ∈ N, is an orthonormal basis of eigenvectors of T , with corresponding eigenvalues λi , i ∈ N, then we can define the self-adjoint operators |T |, |T |1/2 and Ψ as the closures of the linear extensions of the p operators defined by |T |(ei ) = |λi |ei , |T |1/2 (ei ) = |λi |ei and Ψ(ei ) = sgn(λi )ei , i ∈ N. Here the operators |T | and |T |1/2 have compact resolvent and Ψ|T | = T . In addition Ψ commutes with T, |T | and |T |1/2 . Denote by N0 the null space of T and by H1 the orthogonal complement of N0 in L2 [0, 1], then, see [30, page 244], R(|T |) = H1 and |T |1/2 in a self-adjoint operator in H1 with domain D(|T |1/2 ) ∩ H1 . Denote T restricted to H1 1 1 as T˜, then |T˜| 2 is a self-adjoint operator with domain D(|T | 2 ∩ H1 ). Let H2 = L2 [0, 1] and H := H1 ⊕ H2 . Define J : H → H as   ˜ 0 Ψ , J= 0 I ˜ = Ψ|H , and where Ψ 1  E=

0 |T |1/2 |T˜|1/2 −S

 .

as an operator in H with domain D(E) := (H1 ∩ D(|T |1/2 )) ⊕ D(|T |1/2 ). Here J is a unitary operator in H and E is self-adjoint with compact resolvent. Now [x, y] := (Jx, y) defines a Pontryagin space inner product on H. As |J| = I the topology on H considered as a Pontryagin space coincides with its existing Hilbert space topology and the operator J −1 E is J-self-adjoint with compact resolvent. From [4, page 227] there is an almost J-orthonormal Riesz basis for H composed of eigenvectors of J −1 E. Lemma 5.1 If Zj ∈ D(T ), j = 1, . . . , n are linearly independent eigenvectors of the operator pencil W (λ) with eigenvalue λ 6= 0, then λ is an eigenvalue of J −1 E in H and  1 1/2  |T˜| ΨZj λ Vj = , j = 1, . . . , n, Zj are linearly independent eigenvectors. If Zj ∈ D(T ), j = 1, . . . , n are eigenvectors of the operator pencil W (λ), with eigenvalue λ = 0, then 0 is an eigenvalue of J −1 E in H and   |T˜|−1/2 SZj Vj = , j = 1, . . . , n, Zj are linearly independent eigenvectors. In addition, if Zj is at the base of a Jordan chain of length ν for W (λ), then Vj is at the base of a Jordan chain length at least ν for J −1 E.

12

Proof: Firstly, if Z is an eigenvector of W (λ) with non-zero eigenvalue λ, then Z ∈ D(T˜)   U and immediately U := |T˜|1/2 ΨZ ∈ D(|T |1/2 ). Thus is in D(E). Now λZ  (E − λJ)

U λZ



 =

λ|T |1/2 Z − λΨ|T˜|1/2 ΨZ |T˜|ΨZ − λSZ − λ2 Z



 =

λ|T |1/2 Z − λ|T˜|1/2 Z W (λ)Z

 = 0.

Clearly V1 , . . . , Vn are linearly independent, thus we have shown that V1 , . . . , Vn are a linearly independent family of eigenvectors of J −1 E for the eigenvalue λ. Now suppose that Z is at the base of Jordan chain Z = Z [0] , Z [1] , . . . , Z [ν−1] of W (λ) at the eigenvalue λ. Here 0 = T Z [0] − λSZ [0] − λ2 Z [0] , 0 = TZ

[1]

− λSZ

[k]

[1] [k]

2

[1]

2

[k]

−λ Z

0 = T Z − λSZ − λ Z  [k]  U [k] Let V = where Z [k] U [0] = U [k] =

(5.1)

− SZ

[0]

− SZ

[0]

− 2λZ ,

[k−1]

− 2λZ

1 Ψ|T |1/2 Z [0] , λ 1 (Ψ|T |1/2 Z [k] − U [k−1] ), λ

(5.2)

[k−1]

−Z

[k−2]

, 2 ≤ k ≤ ν − 1. (5.3)

1 ≤ k ≤ ν − 1.

We have already shown that (E − λJ)V [0] = 0, so it remains only to show that (E − λJ)V [j] = JV [j−1] , 1 ≤ j ≤ ν − 1,

(5.4)

which is equilavent to ˜ [j−1] = −λΨU ˜ [j] + |T |1/2 Z [j] , ΨU Z [j−1] = |T˜|1/2 U [j] − (S + λI)Z [j] ,

(5.5) (5.6)

for 1 ≤ j ≤ ν − 1. For this we proceed inductively on 1 ≤ j ≤ ν − 1. By the definition of U [j] , (5.5) is obeyed. For j = 1, |T˜|1/2 U [1] − (S + λI)Z [1] = = = = =

1 ˜ 1/2 |T | (Ψ|T |1/2 Z [1] − U [0] ) − (S + λI)Z [1] λ 1 1 W (λ)Z [1] − 2 T Z [0] λ λ 1 1 [1] W (λ)Z − (S + λI)Z [0] λ λ 1 [1] [W (λ)Z − (S + 2λI)Z [0] ] + Z [0] λ Z [0] .

13

Proving the proposition for j = 1. Assuming the proposition true for 1, . . . , j − 1 < ν − 1 we have (as j ≥ 2), 1 ˜ 1/2 |T | (Ψ|T |1/2 Z [j] − U [j−1] ) − (S + λI)Z [j] λ 1 = [W (λ)Z [j] − Z [j−2] − (S + λI)Z [j−1] ] λ = Z [j−1] .

|T˜|1/2 U [j] − (S + λI)Z [j] =

Thus proving the Lemma for the case of λ 6= 0. For λ = 0, one can proceed as above with U [0] = |T˜|−1/2 SZ [0] , U [k] = |T˜|−1/2 (Z [k−1] + SZ [k] ),

1 ≤ k ≤ ν − 1.

For k = 0, EV [0] = 0 since |T |1/2 Z [0] = 0 and |T˜|[1/2] U [0] − SZ [0] = 0. For 1 ≤ k ≤ ν − 1, by definition of U [k] we have that (5.6) holds. Applying |T˜|−1/2 to the Jordan chain condition 0 = Ψ|T |Z [k] − SZ [k−1] − Z [k−2] gives 0 = Ψ|T |1/2 Z [k] − |T˜|−1/2 (SZ [k−1] + Z [k−2] ) = Ψ|T |1/2 Z [k] − U [k−1] , from which (5.5) follows directly. 

 Uj , j = 1, . . . , n, are linearly independent eigenvectors of Zj J −1 E to the eigenvalue λ, then Zj ∈ D(T ), W (λ)Zj = 0 and Z1 , . . . , Zn are linearly  [k]  U [k] , k = 0, . . . , ν − 1, is a Jordan chain of independent. In addition, if V = Z [k] length ν for J −1 E at λ then Z [0] , . . . , Z [ν−1] is a Jordan chain of length at least ν for W (λ) at λ.

Lemma 5.2 If Vj =

Proof: As (J −1 E − λ)Vj = 0 so



Uj Zj



1

1

∈ D(E) but Ψ|T | 2 Zj ∈ D(|T | 2 ) giving Zj ∈

D(T ) and ˜ |1/2 Zj Ψ|T

= λUj ,

(5.7)

|T˜|1/2 Uj − SZj

= λZj .

(5.8)

If λ = 0, then from (5.7), Ψ|T |1/2 Zj = 0 when when operated on by |T |1/2 gives 0 = T Zj = W (0)Zj making Zj an eigenvector of W (λ) at λ = 0. For λ 6= 0, multiply (5.8) by λ and apply |T |1/2 to (5.7). Addition of the resulting equations gives T Zj − λSZj = λ2 Zj

14

showing that Z1 , . . . , Zn are eigenfunctions of W (λ). In order to show the linear independence of Z1 , . . . , Zn suppose α1 Z1 + . . . + αn Zn = 0. Then α1 SZ1 + . . . + αn SZn = 0. So from (5.8), |T˜|1/2 (α1 U1 + . . . + αn Un ) = 0. As T˜ one-to-one we now have α1 U1 + . . . + αn Un = 0. Thus α1 V1 + . . . + αn Vn = 0, and since V1 , . . . , Vn , linearly independent, α1 = . . . = αn = 0, proving the linear independence of Z1 , . . . , Z n . Let V [0] , . . . , V [ν−1] be a Jordan chain to J −1 E at λ. We have already accertained that Z [0] is an eigenvector of W (λ) at λ. Hence Z [0] obeys (5.1). We now show that (5.2) is obeyed. As (5.4) is obeyed for j = 1 we have that (5.5) and (5.6) are obeyed for j = 1. Summing λ times (5.6) with Ψ|T |1/2 applied to (5.5) gives λZ [j−1] + |T |1/2 U [j−1] = W (λ)Z [j] .

(5.9)

Since (E − λJ)V [0] = 0 it follows that |T˜|1/2 U [0] = (S + λI)Z [0] , which when substitued into (5.9) yields (5.2). If j ≥ 2, then summing λ times (5.6) with Ψ|T |1/2 applied to (5.5) and (5.5) with j replaced by j − 1 gives (5.3). Thus concluding the proof. Combining the above two lemmas gives that the eigenvalues of J −1 E and W (λ) coincide and moreover they have the same geometric and algebraic multiplicities. In addition they have identical Jordan structures at each eigenvalue. 

 gn Definition 5.3 If , n ∈ N, is a basis for H ⊕ H, then we say that the sequence fn (fn ) ⊂ H is doubly complete in H.   gn If , n ∈ N, is a basis for H ⊕ (H ⊕ N ) where N is finite dimensional, then we fn say that the sequence (fn ) ⊂ H is almost doubly complete in H ⊕ N . Theorem 5.4 The eigenvectors and associated vectors of the quadratic pencil W (λ) are almost doubly complete in L2 [0, 1]. Proof: As noted  at the beginning of the section, the eigenvectors and their associated U vectors, , of J −1 E form a Riesz basis for H = H1 ⊕ H2 = H1 ⊕ (H1 ⊕ N0 ). Z Here the Z-components are eigenvectors / associated vectors of the pencil W (λ). Thus the eigenvectors and their associated vectors of W (λ) are almost doubly complete in L2 [0, 1] = H1 ⊕ N0 . It should be noted that the eigenvectors and their associated vectors for non-zero eigenvalues of W (λ) are doubly complete in H1 ⊕ H1 .

15

6

Application

In this section we apply the above developed theory to quadratic pencils of SturmLiouville operators on graphs. For general references to Sturm-Liouville problems on graphs we refer the reader to [3, 12, 13, 15, 19, 20, 21, 22] and the bibliographies thereof. An inverse problem for a particular quadratic pencil on trees with ‘generalized Kirchoff boundary conditions’ at each internal vertex was considered in [32]. Let G be an oriented graph with finitely many edges, say K, each of finite length and having the path-length metric. We consider the second-order differential equation with quadratic dependence on the spectral parameter −

d2 y + q(x)y − λs(x)y = λ2 y, dx2

(6.1)

on G, where q and s are real valued and essentially bounded on G. By this we mean the system of equations −

d2 yi + qi (x)yi − λsi (x)yi = λ2 yi , dx2

x ∈ [0, li ], i = 1, . . . , K,

(6.2)

where qi , si (x) and yi denote q|ei , s|ei and y|ei . Here each qi and si is real valued and essentially bounded. Denote by s the minimum (over i) of the essential infima of s˜i (x), x ∈ ei and by s the maximum, over i, of the essential suprema of s˜i (x), x ∈ ei . Here ei will be identified with an interval [0, li ]. In a manner similar to that presented in [10] the differential equations (6.2) may be rewritten as a second order system on [0, 1]. That is, −(M Y 0 )0 + Q(x)Y − λS(x)Y = λ2 Y, (6.3) where Q(x) is a diagonal matrix dependent on the potential on each edge of the  graph,  l −2 −2 s1 (x), . . . , s˜2K (x)). Here s˜j (x) = sj 2j (x + 1) M = 4diag(l1 , . . . , l2K ), and S(x) = diag(˜   l for j = 1, . . . , K and s˜j (x) = sj 2j (1 − x) for j = K + 1, . . . , 2K. At the vertices or nodes of G we impose formally self-adjoint boundary conditions, see [7] for more details regarding the self-adjointness of boundary conditions. In the system formulation, the boundary conditions can be expressed by A∗ Y (0) − B ∗ M Y 0 (0) = 0, ∗



0

Γ Y (1) − ∆ M Y (1) = 0,

(6.4) (6.5)

where A, B, Γ, ∆ are 2K × 2K matrices which obey the following identities in order to ensure self-adjointness, A∗ A + B ∗ B = I, Γ∗ Γ + ∆∗ ∆ = I, A∗ B = B ∗ A and Γ∗ ∆ = ∆∗ Γ. If (τn ) is as given in Section 3 then from [11], τn obeys the following asymptotic estimate as n → ∞, n2 π 2 τn = + O(n), (6.6) L2

16

where L =

P2K

i=1 li

is the total length of the graph.

It should be noted that in [11] the boundary conditions are assumed to be co-normal / elliptic boundary conditions, see [11, Definition]. Thus in order to make use of the asymptotics (6.6) we first need to show that it is possible to remove the constraint of co-normal / elliptic boundary conditions meaning that (6.6) would hold for general self-adjoint boundary conditions at the nodes, as required. We take L2 (G), Hm (G), Hom (G) and the inner product on Hm (G), denoted by (·, ·)m to be as defined in [11]. Without loss of generality, we may assume the boundary conditions at the nodes to take the form K X [αij yj (0) + γij yj (lj )] = 0,

i = 1, . . . , J,

(6.7)

j=1 K X

[αij yj (0) + βij yj0 (0) + γij yj (lj ) + δij yj0 (lj )] = 0,

i = J + 1, . . . , 2K. (6.8)

j=1

Here all possible Dirichlet-like terms are in (6.7), i.e. if (6.8) is written in matrix form then Gauss-Jordan reduction will not allow any pure Dirichlet conditions (i.e. conditions not involving derviatives) linearly independent of (6.7) to be extracted. Let F (x, y) be the sesquilinear from given by Z F (x, y) := (x0 y¯0 + xq y¯) dt + X T N Y¯ , G

where N is a Hermitian symmetric, 2K × 2K, matrix,    x1 (0) y1 (0)  ...   ...     xK (0)   yK (0)   X=  x1 (l1 )  and Y =  y1 (l1 )     ...   ... xK (lK ) yK (lK )

    ,   

with domain D(F ) = {y ∈ H1 (G) | y obeying (6.7)}. Let ρ be the least eigenvalue of N and let Z F− (x, y) := (x0 y¯0 + xq y¯) dt + X T ρI Y¯ . G

Then since ρ is the minimal eigenvalue of N we have that ρI ≤ N, giving min F− (x, x) ≤ min F (x, x) ≤

x∈H1 (G)

x∈D(F )

17

min F (x, x).

x∈H01 (G)

Now Z

0

F− (x, y) − λ(x, y) =

Z

x¯ y dσ + ∂G

x(−¯ y 00 + q y¯ − λy) dt + X T ρI Y¯ .

G

The minimization of F− (x, x) over the elements of H1 (G) on the unit sphere in L2 (G) gives the least eigenvalue of −y 00 +qy = λy on G with the boundary conditions generated by requiring Z 0 = x¯ y 0 dσ + X T ρI Y¯ ∂G

=

K X

[xi (li )¯ yi0 (li )



xi (0)¯ yi0 (0)]

+

i=1

K X

[ρxi (0)¯ yi (0) + ρxi (li )¯ yi (li )]

i=1

for all x ∈ H2 (G). That is y obeys the boundary conditions yi0 (li ) = −ρyi (li ), yi0 (0)

= ρyi (0),

i = 1, . . . , K, i = 1, . . . , K.

The eigenvalue asymptotics for this boundary value problem are well known and are given by n2 π 2 λi,− + O(1), i = 1, . . . , K, n = 0, 1, 2, . . . . n = li2 The remainder of [11] now holds giving (6.6) for general self-adjoint boundary conditions. Thus, from section 4, we obtain that for large modulus eigenvalues of the boundary value problem given by (6.3)-(6.5) when repeated according multiplicity take on the asymptotic form nπ + O(1), n → ∞, (6.9) λ± n =± L P where L = 2K i=1 li is the total length of the graph. Also all but finitely many of the eigenvalues are real and semi-simple. From section 5 it follows that the eigenfunctions and associated functions of the quadratic eigenvalue problem on a graph are almost doubly complete in L2 (G).

References [1] F.V. Atkinson, Discrete and continuous boundary value problems, Academic Press, London, 1964. [2] F.V. Atkinson, H. Langer, R. Mennicken, Sturm-Liouville problems with coefficients which depend analytically on the eigenvalue parameter, Acta Sci. Math. (Szeged), 57 (1993), 25-44.

18

[3] J. Avron, Adiabatic quantum transport in multiply connected systems, Reviews of Modern Physics., 60 (1988), 873-915. [4] T.Y. Azizov, I.S. Iokhvidov, Linear operators in spaces with an in metric, WileyInterscience , 1989. [5] P.A. Binding, On generalised and quadratic eigenvalue problems, Applicable Analysis, 12 (1981), 27-45. [6] P.J. Browne, B.A. Watson, Oscillation theory for a quadratic eigenvalue problem, Quaestiones Mathematicae, 31 (2008), 345-357. [7] R. Carlson, Adjoint and self-adjoint differential operators on graphs, Electronic J. Differential Equations, 1998 (1998), No. 06, 1-10. [8] E. A. Coddington, N. Levinson, Theory of ordinary differential equations , McGraw-Hill, New York, 1955. [9] S. Currie, Spectral theory of differential operators on graphs, PhD Thesis, University of the Witwatersrand, Johannesburg, 2006. [10] S. Currie, B.A. Watson, Eigenvalue asymptotics for differential operators on graphs, J. Com. Appl. Math., 182 (2005), 13-31. [11] S. Currie, B.A. Watson, Dirichlet-Neumann bracketing for boundary-value problems on graphs, Elec. J. Diff. Eq, 2005 (2005), 1-11. [12] P. Exner, M. Helm, P. Stollmann, Localization on a quantum graph with a random potential on the edges , Rev. Math. Phys., 19 (2007), 923-939. ˇ ˇt ˇovic ˇek, Quantum interference on graphs controlled by [13] P. Exner, P. Seba, P. S an external electric field , J. Phys., A 21 (1988) no. 21, 4009-4019. [14] M.G. Gasymov, G.S. Guseinov, Determination of diffusion operator on spectral data, Dokl. Akad. Nauk. Azerb. SSR, 37 (1981), 19-23. [15] N. Gerasimenko, B. Pavlov, Scattering problems on non-compact graphs, Theoretical and Mathematical Physics, 74 (1988), 230-240. [16] G. SH. Guseinov, On spectral analysis of a quadratic pencil of Sturm-Liouville operators, Soviet Math. Dokl., 32 (1985), 859-862. [17] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. [18] H. Koyunbakan, E.S. Panakhov, Half-inverse problem for diffusion operators on the finite interval, J. Math. Anal. Appl., 326 (2007), 1024-1030. [19] V. Kostrykin, R. Schrader, Quantum wires with magnetic fluxes, Commun. Math. Phys., 237 (2003), 161-179. [20] T. Kottos, U. Smilansky, Chaotic scattering on graphs, Phys. Rev. Lett, 85 (2000), 968-971. [21] P. Kuchment, Graph models for waves in thin structures, Waves Random Media, 12 (2002), R1-R24.

19

[22] P. Kuchment, Differential and psuedo-differential operators on graphs as models of mesoscopic systems, Analysis and Applications ed H. Begehr, R. Gilbert, M.W. Wang (Dordrecht: Kluwer Academic), (2003), 7-30. [23] P. Lancaster, A. Sckalikov, Damped vibrations of beams and related spectral problems, Canadian Applied Mathematics Quarterly, 2 (1994), 45-90. [24] M. A. Naimark, Linear differential operators, part I, Frederick Ungar Publishing Co., New York, 1967. [25] B. Najman, Eigenvalues of the Klein-Gordon equation, Proc. Edin. Math. Soc., 26 (1983), 181-190. [26] R.G.D. Richardson, Theorems of oscillation for two linear differential equations of the second order with two parameters, Trans. of the American Math. Soc., 13 (1912), 22-34. [27] G.F. Roach, B.D. Sleeman, On the spectral theory of operator bundles, Applicable Analysis, 7 (1977), 1-14. [28] G.F. Roach, B.D. Sleeman, On the spectral theory of operator bundles II, Applicable Analysis, 9 (1979), 29-36. [29] A.A. Shkalikov, Boundary value problems for ordinary differential equations with parameter in the boundary conditions, Trudy Seminara imeni I.G. Petrovskogo, 9 (1983), 190-229. [30] A.E. Taylor, D.C. Lay, Introduction to functional analysis, 2nd edition, Robert E. Krieger, 1986. [31] J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, 1980. [32] V. Yurko, Recovering differential pencils on compact graphs, J. Diff. Eq., 244 (2008), 431-443.

20

Quadratic eigenvalue problems for second order systems

We consider the spectral structure of a quadratic second order system ...... [6] P.J. Browne, B.A. Watson, Oscillation theory for a quadratic eigenvalue prob-.

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