Nov 10, 2006 - the set of eigenfunctions of the non-standard two-parameter eigenvalue problem, λyâ²â²(s) + µa(s)y(s)
Completeness theorems for a non-standard two-parameter eigenvalue problem
∗
Melvin Faierman School of Mathematics The University of New South Wales NSWU, Sydney NSW 2052, Australia
Manfred M¨oller and Bruce A. Watson
†
School of Mathematics University of the Witwatersrand Private Bag 3, P O WITS 2050, South Africa November 10, 2006
Abstract We consider two simultaneous Sturm-Liouville systems coupled by two spectral parameters. However, unlike the standard multiparameter problem, we now suppose that the principal part of each of the differential operators is multiplied by a different parameter. In a recent paper, Faierman and Mennicken derived various results concerning the ∗
Keywords: Multiparameter eigenvalue problems, completeness, Sturm-Liouville.
Mathematics subject classification (2000): 34B08, 35G15, 35P10. † MM and BAW supported in part by the John Knopfmacher Centre for Applicable Analysis and Number Theory.
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eigenvalues and eigenfunctions, and in particular, they established the oscillation theory for this system. Here we continue this investigation focusing on the completeness of the set of eigenfunctions in a suitable function space. If either one of the potentials is identically zero, the completeness of the eigenfunctions is established, whereas, if this condition fails, then we show the existence of an essential spectrum having non-zero points. The completeness problem for this latter case will be left for a later work.
1
Introduction
The object of this paper is to show the completeness, in a suitable function spaces, of the set of eigenfunctions of the non-standard two-parameter eigenvalue problem, λy ′′ (s) + µa(s)y(s) + b(s)y(s) = 0,
0 ≤ s ≤ 1,
′
= d/ds,
y(0) = y(1) = 0, −λc(t)z(t) + µz ′′ (t) + d(t)z(t) = 0,
(1.1) (1.2)
0 ≤ t ≤ 1,
z(0) = z(1) = 0,
′
= d/dt,
(1.3) (1.4)
where a, b, c, and d are real-valued continuous functions, and, in addition, a and c are positive. By an eigenvalue of the system (1.1)-(1.4) we mean a tuple (λ, µ) of complex numbers, not both zero, such that (1.1) has a non-trivial solution, say y(s, λ, µ), satisfying (1.2) and such that (1.3) has a non-trivial solution, say z(t, λ, µ), satisfying (1.4). The product y(s, λ, µ)z(t, λ, µ) =: y⊗z(s, t, λ, µ) is called an eigenfunction of (1.1)-(1.4) corresponding to the eigenvalue (λ, µ). In [5], the reality of the eigenvalues, the orthogonality of the eigenfunctions, and an analogy with the oscillation theory of Klein for (1.1)-(1.4) were proved. Here the focus is on the completeness of the eigenfunctions associated with (1.1)-(1.4). The approach used is that proposed by Hilbert [6], i.e., reduction of (1.1)-(1.4) to a boundary problem
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for a partial differential equation. It should be noted that alternative approaches to multiparameter spectral theory can be found in [3, 12]. In particular, there is associated with the boundary problem a linear operator pencil S(λ) = λA − T acting in a certain Hilbert space, where A is a selfadjoint partial differential operator, T is a symmetric partial differential operator and the order of A exceeds that of T . Standard multiparameter spectral theory only considers cases when the operator A is elliptic and is associated with a coercive sesquilinear form. Consequently regularity results are available which ensure the completeness of the set of eigenvectors of S(λ) corresponding to its non-zero eigenvalues and that these eigenvectors coincide with the eigenfunctions of the multiparameter system. Our proof of the completeness of the set of eigenfunctions relies on the first component of every eigenvalue of the system (1.1)-(1.4) being a non-zero eigenvalue of S(λ). In the problem at hand, A is non-elliptic and the corresponding sesquilinear form non-coercive, making the above approach inapplicable, see the open questions at the end of [5]. In this paper we suppose that at least one of the functions b(s) and d(t) has no zeros. Then, by means of the partial differential equation approach, we establish a regularity result which allows us to resolve the completeness problem for the case of b(s) ≡ 0 and for the case of d(t) ≡ 0. However, when both b(s) and d(t) are not identically zero, the problem remains open. For the case b(s) ≡ 0, we show that the only point in the essential spectrum of S(λ) is λ = 0. The remaining spectrum of S(λ) consists solely of eigenvalues of finite multiplicity. Standard techniques can thus be applied to establish the completeness theorem for the eigenvectors of S(λ) corresponding to its non-zero eigenvalues. These eigenvectors and eigenvalues can now be identified with the eigenfunctions and first components of the eigenvalues, respectively, of the system (1.1)-(1.4). We observe that for d(t) ≡ 0 by replacing λ with µ, µ with −λ, b with d, d with −b and interchanging a and c as well as the independent variables s and t this case can be reduced to that of b(s) ≡ 0.
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For the case where both b(s) and d(t) do not vanish identically, we show that part of the essential spectrum of S(λ) lies in R\{0}. Thus there may be non-zero eigenvalues of S(λ) embedded in the essential spectrum. Hence, prior to considering the completeness of the eigenfunctions of (1.1)-(1.4), we must locate the non-zero eigenvalues of S(λ), if any, which are embedded in the essential spectrum. Bearing in mind that each such eigenvalue could be a limit point of the spectrum of S(λ), we must next ascertain precisely what function spaces are spanned by the eigenvectors of S(λ) corresponding to both embedded eigenvalues and non-embedded eigenvalues. Finally it remains to identify these eigenvectors with the eigenfunctions of the system (1.1)-(1.4). At present the partial differential equation approach has not yielded all the required information, and hence the question of the completeness remains open in this case. In Section 2, results from [5], which are referred to in the sequel, are given. The operator A is defined in Section 3. The regularity of functions in the domain of A is considered in Section 4. The completeness theorem for the cases b(s) ≡ 0 or d(t) ≡ 0 is proved in Section 5. In Section 6 cases when these latter conditions on b(s) and d(t) do not hold are considered. In particular it is shown that the non-zero portion of the essential spectrum of S(λ) is not empty.
2
Preliminaries
In this section we define the operator A of the Introduction and summarise from [5] the necessary spectral theory for (1.1)-(1.4). In order to avoid pathological cases in which the system has eigenvalues of infinite multiplicity, we make the following assumption.
Assumption 2.1 It will henceforth be assumed that at least one of the functions b(s) and d(t) has no zeros on its respective interval of definition, and if b(s) 6≡ 0 (resp. d(t) 6≡ 0), then b(s) (resp. d(t)) is not a scalar multiple of a(s) (resp. c(t)).
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Assumption 2.1 together with (1.1) and (1.3) ensure that if (λ, µ) is an eigenvalue of the system (1.1)-(1.4), then neither λ nor µ is zero. From [5] we have the following result.
Proposition 2.2 The eigenvalues of (1.1)-(1.4) are real.
Let Ω be the open square (0, 1) × (0, 1) in the (s, t)-plane. Then Proposition 2.4 from [5] gives:
Proposition 2.3 Let {(λj , µj )}k1 , k ∈ N, denote k distinct eigenvalues of (1.1)-(1.4) and let {yj ⊗ zj }k1 denote the corresponding set of eigenfunctions. Then yj ⊗ zj , j = 1, . . . , k form a linearly independent set in L2 (Ω).
In order to be more precise about the eigenfunctions of (1.1)-(1.4), let us introduce the following notation. For any pair of scalars (λ, µ) let y˜(s, λ, µ) denote the solution of (1.1) satisfying y˜(0, λ, µ) = 0, y˜′ (0, λ, µ) = 1. Similarly let z˜(t, λ, µ) denote the solution of (1.3) satisfying z˜(0, λ, µ) = 0, z˜′ (0, λ, µ) = 1.
Note that y˜(s, λ, µ) and z˜(t, λ, µ) are of class C 2 with respect to s and t respectively and that if (λ, µ) is an eigenvalue of (1.1)-(1.4), then any corresponding eigenfunction is a scalar multiple of y˜(s, λ, µ)˜ z (t, λ, µ) = (˜ y ⊗ z˜)(s, t, λ, µ). The following oscillation theorem was proved in [5].
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Theorem 2.4 Let (m, n) be any tuple of non-negative integers. There exists at least 1 and at most 4 distinct eigenvalues of (1.1)-(1.4), say {(λj , µj )}k1 , 1 ≤ k ≤ 4, such that for each j the function y˜(s, λj , µj ) has precisely m zeros in 0 < s < 1, and the function z˜(t, λj , µj ) has precisely n zeros in 0 < t < 1.
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The operator A
The results of the previous section give that the eigenfunctions of (1.1)-(1.4) form a countably infinite linearly independent set in L2 (Ω). The question now arises of whether this set is complete in L2 (Ω). We approach this question via studying an associated partial differential equation boundary value problem, as proposed by Hilbert [6, pp. 262–266]. This in turn gives rise to the operator A, the topic of this section. Let (λ, µ) be an eigenvalue of (1.1)-(1.4) and w(s, t) = y˜(s, λ, µ)˜ z (t, λ, µ). From (1.1)(1.4) it follows that w is a solution of the boundary problem λ(D12 D22 u + acu) + (bD22 u − adu) = 0 in
Ω,
(3.1)
u = 0 on
Γ,
(3.2)
where D1 = ∂/∂s, D2 = ∂/∂t, and Γ denotes the boundary of Ω. It also follows from (1.1)-(1.4) that w(s, t) is a solution of the boundary problem (3.1)′ , (3.2), where (3.1)′ is obtained from (3.1) by replacing λ by µ and bD22 u − adu by dD12 u + bcu . In this paper, our focus will be on (3.1)-(3.2). Remark The differential equation (3.1) is not elliptic. Hence standard multiparameter methods (cf. [4, Chapter 2]) for obtaining regularity of the solutions of (3.1) no longer apply. The boundary problem (3.1)-(3.2) can be expressed as an operator eigenvalue problem by means of a sesquilinear form (cf. [4, Chapter 2]).
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Let (·, ·) and k · k denote the inner product and norm in L2 (Ω), respectively. For m a non-negative integer, denote by H m (Ω) denote the Sobolev space of order m related to ˚m (Ω) be the closure of C ∞ (Ω) in H m (Ω). L2 (Ω) with norm denoted by k · km,Ω . Let H 0 Let ˚1 (Ω) | D1 D2 u ∈ L2 (Ω)}. V := {u ∈ H If u ∈ V , then the restriction of u to Γ is well defined, i.e. trΓ u exists on Γ and is 0, where trΓ denotes trace of u on Γ (see [11, p. 176]). The following result was proved in [5].
Proposition 3.1 Let u ∈ V . Then trΓ D1 u exists and is 0 on those portions of Γ for which t = 0 and t = 1. Similarly, trΓ D2 u exists and is 0 on those portions of Γ for which s = 0 and s = 1.
For u, v ∈ V define the sesquilinear form B(u, v) by B(u, v) := (D1 D2 u, D1 D2 v) + (acu, v). Let γ0 = inf {a(s)c(t)}. Ω
Then γ0 > 0, and the following theorem holds, see [5, Proposition 4.3].
Proposition 3.2 The form B is closed symmetric and densely defined in L2 (Ω) with γ0 as a lower bound. Furthermore, there is a constant C > 0 for which B(u, u) ≥ Ckuk21,Ω
for all
u ∈ V.
Observe that V equipped with the inner product h·, ·i = B(·, ·) is a Hilbert space. We shall henceforth consider V as this Hilbert space and let k · kV = h·, ·i1/2 denote the associated norm on V . Denote by A the self-adjoint operator in L2 (Ω) associated with B (see [9, p. 323]) and let its domain be denoted by D(A). Here A ≥ γ0 , A−1 exists and is bounded.
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4
Regularity
Since the operator A is not elliptic, currently published results do not determine the regularity of functions in D(A). The regularity of the elements of D(A) is thus the subject of this section. Proposition 4.1 Let u ∈ L2 (Ω). Then u ∈ D(A) if and only if u ∈ H 2 (Ω), D1 D22 u, D2 D12 u, D12 D22 u ∈ L2 (Ω) and trΓ u = 0. In this case, Au = D12 D22 u + acu. Proof: We will make use of the fact that u ∈ D(A) if and only if there is w ∈ L2 (Ω) such that (w, v) = B(u, v) for all v ∈ V (see [9, Theorem VI.2.1]). In this case, Au = w. First let u have the stated representation. Then u ∈ H 2 (Ω) and the trace condition for u imply u ∈ V . For all v ∈ V we have (D12 D22 u, v) = −(D1 D22 u, D1 v) = (D1 D2 u, D1 D2 v), where we have used Nikod´ ym’s theorem (see [11, pp. 73 and 179]) and Proposition 3.1. Thus (D12 D22 u + acu, v) = (D1 D2 u, D1 D2 v) + (acu, v) = B(u, v) for all v ∈ D(V ), and the assertions u ∈ D(A) as well as Au = D12 D22 u + acu follow. Conversely, let u ∈ D(A) and let D ′ (Ω) be the space of distributions on Ω. Let h·, ·iΩ denote the bilinear form in the dual pair (D ′ (Ω), C0∞ (Ω)). Since C0∞ (Ω) ⊂ V , it follows for w as in the first paragraph of this proof and all v ∈ C0∞ (Ω) that hw, viΩ = (w, v) = B(u, v) = (D1 D2 u, D1 D2 v) + (acu, v) = hD1 D2 u, D1 D2 viΩ + hacu, viΩ = hD12 D22 u, viΩ + hacu, viΩ ,
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which means that w = D12 D22 u + acu
(4.1)
in the sense of distributions. Note that g := w − acu ∈ L2 (Ω).
(4.2)
By definition of V , u ∈ V implies h := D1 D2 u ∈ L2 (Ω). Together with (4.1) and (4.2) this leads to D1 D2 h = g. Putting ˜ t) = h(s,
Z tZ 0
s
g(σ, τ )dσ dτ,
0
it follows that ˜ = 0, D1 D2 (h − h) and thus ˜ =1⊗h ˜2 D2 (h − h) ˜ 2 ∈ D ′ (0, 1) (see [8, Theorem 3.1.4′ ]). Here (f1 ⊗ f2 )(s, t) = f1 (s)f2 (t). Since with h differentiation is surjective in D ′ (0, 1) (see [7, Theorem 3.6.4 and Corollary 3.6.1]), there is h2 ∈ D ′ (0, 1) such that h′2 = ˜ h2 . Thus ˜ − 1 ⊗ h2 ) = 0, D2 (h − h and repeating the above argument this implies ˜ − 1 ⊗ h2 = h1 ⊗ 1 h−h with h1 ∈ D ′ (0, 1). So ˜ + h1 ⊗ 1 + 1 ⊗ h2 . h=h ˜ ∈ L2 (Ω), from which it follows that Clearly, h h1 ⊗ 1 + 1 ⊗ h2 ∈ L2 (Ω),
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and hence by Fubini’s theorem, s 7→ h1 (s) + h2 (t), t 7→ h1 (s) + h2 (t) belong to L2 (0, 1) for almost all t and s, respectively. Thus h1 , h2 ∈ L2 (0, 1). Repeating what we deduced from D1 D2 h = g for D1 D2 u = h, it follows that u = f + u3 ⊗ 1 + 1 ⊗ u 4 ,
(4.3)
where u3 , u4 ∈ L2 (0, 1) and f (s, t) =
Z tZ 0
s
h(σ, τ ) dσ dτ. 0
Putting u0 (s, t) = u1 (s) =
Z
s Z t Z s1
Z0 s
0
0
Z
t1
g(s2 , t2 ) dt2 ds2 dt1 ds1 ,
(4.4)
0
h1 (σ) dσ,
0
u2 (t) =
Z
t
h2 (τ ) dτ,
0
(4.3) leads to u = u 0 + u 1 ⊗ t + s ⊗ u 2 + u3 ⊗ 1 + 1 ⊗ u 4 ,
(4.5)
and the trace condition for u holds since u ∈ V . We still have to prove the regularity properties for u stated in the proposition. This is immediately clear for u0 . Now u ∈ H 1 (Ω) implies that D1 u0 + h1 ⊗ t + 1 ⊗ u2 + u′3 ⊗ 1 = D1 u ∈ L2 (Ω), D2 u0 + u1 ⊗ 1 + s ⊗ h2 + 1 ⊗ u′4 = D2 u ∈ L2 (Ω), which shows u3 , u4 ∈ H 1 (0, 1). The boundary condition for u and u0 (s, 0) = 0 = u0 (0, t) as well as u1 (0) = 0 = u2 (0) lead to 0 = u3 (s) + u4 (0), 0 = u3 (0) + u4 (t),
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and thus u3 (s) = −u4 (0), u4 (t) = −u3 (0), which implies u3 ⊗ 1 + 1 ⊗ u4 = 0 and therefore (4.5) simplifies to u = u 0 + u1 ⊗ t + s ⊗ u 2 .
(4.6)
Evaluating the trace at t = 1 and s = 1, we get 0 = u0 (s, 1) + u1 (s) + su2 (1), 0 = u0 (1, t) + tu1 (1) + u2 (t), which leads to u1 (s) = −u0 (s, 1) − su2 (1),
(4.7)
u2 (t) = −u0 (1, t) − tu1 (1).
(4.8)
From the representation of u0 it follows immediately that u0 (·, 1) and u0 (1, ·) belong to H 2 (0, 1). This implies that u1 , u2 ∈ H 2 (0, 1), and thus also u1 ⊗ t and s ⊗ u2 have the required regularity properties.
Corollary 4.2 For f ∈ L2 (Ω) define Z sZ tZ (W0 f )(s, t) = 0
0
s1 0
Z
t1
f (s2 , t2 ) dt2 ds2 dt1 ds1
0
and (W f )(s, t) = (W0 f )(s, t) + st(W0 f )(1, 1) − s(W0 f )(1, t) − t(W0 f )(s, 1). Then W is bounded from L2 (Ω) into D(A) and A−1 = W − W acA−1 .
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(4.9)
Proof: It is obvious that W maps into D(A) (that trΓ W f = 0 is seen by a simple substitution) and thus W is bounded from L2 (Ω) into D(A) by the closed graph theorem. Let w ∈ L2 (Ω) and u = A−1 w. Referring to the proof of Proposition 4.1 for terminology, we observe that (4.7)-(4.8) imply that u1 (1) + u2 (1) = −u0 (1, 1), and tu1 (s) + su2 (t) = −tu0 (s, 1) − su0 (1, t) − st(u1 (1) + u2 (1)) = stu0 (1, 1) − tu0 (s, 1) − su0 (1, t). By (4.4), u0 = W0 g which implies u = W g by (4.6) and the above representation of tu1 (s) + su2 (t). Recalling the definition of g in (4.2) and u = A−1 w, it follows that u = W g = W (w − acu) = W (w − acA−1 w), and (4.9) is established.
Proposition 4.3 Let u ∈ D(A). Then trΓ D12 u exists and is 0 on those portions of Γ for which t = 0 and t = 1. Likewise, trΓ D22 u exists and is zero on those portions of Γ for which s = 0 and s = 1.
Proof: We shall only prove the assertion concerning trΓ0 D12 u on Γ0 , where Γ0 denotes that portion of Γ for which t = 0. The remaining assertions can be proved in a similar manner. We also let φ(s) denote a test function on 0 < s < 1 and let ψ(t) be a function of class C ∞ in −∞ < t < ∞ such that ψ(t) = 1 for |t| ≤ 1/4 and ψ(t) = 0 for |t| ≥ 1/2. We observe from Proposition 4.1 that D12 u, D2 D12 u ∈ L2 (Ω), and hence it follows from Nikod´ ym’s theorem that D12 u may be modified on a set of measure zero so that for almost every s in (0, 1), D12 u is absolutely continuous in 0 ≤ t ≤ 1 and the distributional derivative D2 D12 of D12 u is equal to the classical derivative of D12 u with respect to t almost everywhere in 0 ≤ t ≤ 1. Furthermore, trΓ0 D12 u exists on Γ0 and belongs to L2 (I0 ), where I0 denotes the interval 0 < s < 1, while for almost every s in (0, 1), we
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have γ2 u = − trΓ0 D12 u.
R1 0
D2 (ψD12 u)dt, where for brevity we have written γ2 u for trΓ D12 u|Γ0 =
By arguing again with Nikod´ ym’s theorem we can show that there exists the sequence {tj }∞ 1 where
1 4
> tj > · · · > 0 and tj → 0 as j → ∞ such that for each j, D1 u(·, tj ) and
D12 u(·, tj ) belong to L2 (I0 ), Z Z 1 γ2 uφ ds = 0
1 0
D12 u(s, tj )φ(s) ds
−
Z
Ωj
(D2 D12 u)φ ds dt,
and Z
1 0
D12 u(s, tj )φ(s) ds
=−
Z
1
D1 u(s, tj )D1 φ(s) ds = −
0
Z
(D2 D1 u)D1 φ ds dt, Ωj
where we have made use of Proposition 3.1 and Ωj denotes the rectangle in the (s, t)plane defined by the inequalities 0 < s < 1, 0 < t < tj . Since these equalities hold for R1 every j, we conclude that 0 γ2 uφ ds = 0, and hence, since φ is arbitrary, we must have γ2 u = 0 on Γ0 , which completes the proof of the proposition.
5
Completeness
In this section, we will consider the case in which at least one of b(s) and d(t) is identically zero. Returning to (3.1), let T be the operator acting in L2 (Ω) defined by T u = adu for u in D(T ) := L2 (Ω). From Proposition 4.1, D(A) ⊂ D(T ). Let S(λ), λ ∈ C, denote the operator pencil in L2 (Ω) defined by S(λ)u = (λA − T )u,
for u ∈ D(S) := D(A).
Define the eigenvalues, eigenvectors, and associated vectors of the boundary problem (3.1)-(3.2) as those for the pencil S(λ) (see [10] for terminology concerning pencils). We
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are now in a position to present the completeness theorem associated with the system (1.1)-(1.4). Theorem 5.1 Suppose that either b(s) = 0 in 0 ≤ s ≤ 1 and that a(s) and d(t) are Lipschitz continuous or d(t) = 0 in 0 ≤ t ≤ 1 and b(s) and c(t) are Lipschitz continuous. Then the set of eigenfunctions of the two-parameter system (1.1)-(1.4) is complete in L2 (Ω). Proof: As mentioned in the introduction, it may without loss of generality be assumed that b(s) is identically zero. Thus, from Assumption 2.1, d(t) 6= 0 in 0 ≤ t ≤ 1. ˚ 1 (Ω) if u ∈ V and that Observe that T u ∈ H hA−1 T u, vi = (T u, v),
for
u, v ∈ V.
Consequently, see [9, Theorem VI.2.23, p. 331], K = A−1 T is a compact self-adjoint operator in V . Since S(λ) = A(λ − K) it follows that the eigenvalues of S(λ) are real and that the eigenvectors of S(λ) corresponding to its non-zero eigenvalues form an orthonormal basis in ran K with respect to the inner product h·, ·i, where ran denotes range. Since ⊥
ran K = N (K) = {u ∈ V | T u = 0} = {u ∈ V | d(t)u(·, t) = 0 a. e.} = {0}, the eigenvectors of S(λ) are complete in V and thus also in L2 (Ω). Since the eigenfunctions of (1.1)-(1.4) corresponding to distinct eigenvalues are linearly independent eigenvectors of S(λ), from the previous paragraph we need only show that they constitute all of these eigenvectors. Let λm be a non-zero eigenvalue of S(λ) and w a corresponding eigenvector. Let {µn }∞ 0 denote the sequence of eigenvalues of (1.1)-(1.2) with λ = λm . For brevity write 1/2 .Z 1 2 y˜n (s) = y˜(s, λm , µn ) . a(s)˜ y (s, λm , µn ) ds 0
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The y˜n form an orthonormal basis in L2 (I0 ; a), where I0 is the interval (0, 1). Hence for almost every t in (0, 1) we have w(s, t) =
∞ X
zn (t)˜ yn (s),
(5.1)
n=0
where zn (t) =
Z
1
a(s)w(s, t)˜ yn (s) ds.
0
From (3.1) and Proposition 4.1 observe that for a given n in (5.1), λm
Z
0
1
yn (s) ds + (λm c(t) − d(t))zn (t) = 0 for a.e. t in (0, 1). D12 D22 w(s, t)˜
(5.2)
Proposition 4.1 gives that D22 w, D1 D22 w, and D12 D22 w are in L2 (Ω). Hence Proposition 4.3 and the Nikod´ ym theorem allow us to deduce from (1.1) and (5.2) that −λm c(t)zn (t) + µn
Z
1 0
a(s)D22 w(s, t)˜ yn (s) ds + d(t)zn (t) = 0 for a.e. t in (0, 1).
(5.3)
Consider (5.3). Since w ∈ H 2 (Ω), by the Sobolev embedding theorem (cf. [1, Theorem 3.9, p.32]), w(s, t) can be modified on a set of measure 0 to give w ∈ C 0 (Ω). Hence we conclude, from the definition, that zn ∈ C (0) (I1 ), where I1 = {t ∈ R | 0 < t < 1}. If φ ∈ C0∞ (I1 ) and h·, ·iI1 denotes the bilinear form in the dual pair (D ′ (I1 ), C0∞ (I1 )), then from the theorems of Fubini and Nikod´ ym, Z 1 Z ′ ′ aw(·, t)˜ yn ds, φ = − hzn , φ iI1 = 0
1 0
. aD2 w(·, t)˜ yn ds, φ I1
Hence zn ∈ H 1 (I1 ) ∩ AC(I1 ) with zn′ (t) =
Z
1
aD2 w(·, t)˜ yn ds 0
in the sense of distributions on I1 . Here AC(I1 ) denotes the space of absolutely continym’s theorem that uous functions on I1 . Since D2 w(s, t) ∈ H 1 (Ω) it follows from Nikod´ D2 w(s, t) can be modified on a set of measure zero to give zn ∈ C 1 (I1 ) and Z 1 ′ aD2 w(·, t)˜ yn ds. zn (t) = 0
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Proceeding with hzn′ , φ′ iI1 as we did for hzn , φ′ iI1 above, we obtain zn′ ∈ H 1 (I1 ) ∩ AC(I1 ) with zn′′ (t) =
Z
1
0
aD22 w(·, t)˜ yn ds
in the sense of distributions on I1 . Here the derivative of zn′ coincides with its distributional derivative almost everywhere in I1 . In light of the last result, (1.3)-(1.4) and (5.3) we can modify zn′′ on a set of measure zero so that zn is a C 2 solution of (1.3)-(1.4) with λ = λm and µ = µn . In particular, the non-zero elements in the sequence (˜ yn (s)zn (t)) are linearly independent eigenfunctions of S(λm ). Since the eigenvalues of S(λ) are of finite multiplicity, it now follows that w(s, t) is a finite linear combination of eigenfunctions of (1.1)-(1.4). As w(s, t) was arbitrarily chosen, the proof is complete.
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The case b(s) 6≡ 0 and d(t) 6≡ 0
Here we consider (3.1), with both b(s) and d(t) not identically zero. Let T denote the operator acting in L2 (Ω) defined by T u = adu − bD22 u,
for
u ∈ D(T ).
Here the domain of T , D(T ), is given by D(T ) = {u ∈ L2 (Ω) | D22 u ∈ L2 (Ω)}. From Proposition 4.1 it follows that D(A) ⊂ D(T ). Define the operator pencil S(λ) in L2 (Ω), λ ∈ C, by S(λ)u = (λA − T )u for u in the domain of S(λ), D(S) := D(A). Define the eigenvalues, eigenvectors, and associated vectors of the boundary problem (3.1)-(3.2) to be those of the pencil S(λ)
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(see [10] for terminology concerning pencils). Since none of these quantities changes if S(λ) is replaced by A−1 S(λ) = λI − A−1 T0 , where T0 = T |D(A) , we will henceforth consider A−1 T0 with domain is D(A). The operator A−1 T0 is symmetric in V with respect to the inner product h·, ·i since hA−1 T0 u, vi = (T u, v),
for
u, v ∈ D(A).
Although T has a larger domain, it is more convenient and sufficient to consider the operator T0 . Now let K := A−1 T0 be the closure of A−1 T0 in V , which is a symmetric operator.
Proposition 6.1 Let b(s) 6≡ 0. Then K : V → V is bounded.
Proof: The map A−1/2 : L2 (Ω) → V is an isomorphism, isometric with respect to the inner products (·, ·) on L2 (Ω) and h·, ·i on V . Hence we need only prove that A−1/2 T0 A−1/2 has a bounded closure in L2 (Ω), where this operator is defined on V = A1/2 (D(A)). Write A−1/2 T0 A−1/2 = A−1/2 adA−1/2 − (A−1/2 D2 )bD2 A−1/2 . Here A−1/2 adA−1/2 is bounded in L2 (Ω). Since A−1/2 : L2 (Ω) → V is bounded so is D2 A−1/2 in L2 (Ω). Let Z = {bD2 A−1/2 u : u ∈ V }. Then D2 u ∈ L2 (Ω) for u ∈ Z, and it is sufficient to show that A−1/2 D2 |Z has a bounded closure in L2 (Ω). But from [9, Theorem 5.28, p. 168] we have ∗ ∗∗ ⊂ (D2 |Z )∗ A−1/2 A−1/2 D2 |Z = A−1/2 D2 |Z and (D2 |Z )∗ |V = −D2 |V . Thus (D2 |Z )∗ A−1/2 = −D2 A−1/2
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is bounded and A−1/2 T0 A−1/2 = A−1/2 adA−1/2 − A−1/2 D2 bD2 A−1/2 is a bounded operator in L2 (Ω). From Corollary 4.2 we have K = W T0 − W acK, where W T0 is bounded.
Proposition 6.2 σess (K) = σess (W T0 ).
Proof: The embeddings of D(A) into V and V into H 1 (Ω) are continuous, while the embedding of H 1 (Ω) into L2 (Ω) is compact. The maps K : V → V and W : L2 (Ω) → D(A) are bounded, as is multiplication by ac in L2 (Ω). Hence the theorem follows from [9, Theorem IV.5.35]). Let K0 be the closure in V of −W bD22 |D(A) , where we note that K0 is a bounded operator in V .
Proposition 6.3 σess (K) = σess (K0 ).
Proof: As above, one shows that W ad is compact in V . We now give a representation of W bD22 . Observe that for f ∈ D(A), D22 f ∈ L2 (Ω), and thus t 7→ D22 f (s, t) belongs to L2 (0, 1) for almost all s. Hence we may assume that t 7→ D2 f (s, t) is continuous on [0, 1] for all s. Then, for f ∈ D(A), (W0 bD22 f )(s, t) =
Z
=
Z
=
0
s Z s1
0 s Z s1
Z0 s Z0 s1 0
b(s2 )
Z tZ
b(s2 )
Z
0
0
t1
(D22 f )(s2 , t2 ) dt2 dt1 ds2 ds1
t
[(D2 f )(s2 , t1 ) − (D2 f )(s2 , 0)] dt1 ds2 ds1 0
b(s2 )[f (s2 , t) − t(D2 f )(s2 , 0)] ds2 ds1 .
0
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This leads to (W bD22 f )(s, t) =
Z
0
s Z s1
0
0
b(s2 )[f (s2 , t) − t(D2 f )(s2 , 0)] ds2 ds1 Z 1 Z s1 b(s2 )(D2 f )(s2 , 0) ds2 ds1 −st 0 0 Z 1 Z s1 −s b(s2 )[f (s2 , t) − t(D2 f )(s2 , 0)] ds2 ds1 Z 0s Z 0s1 b(s2 )(D2 f )(s2 , 0) ds2 ds1 +t 0 0 Z 1 Z s1 Z s Z s1 b(s2 )f (s2 , t) ds2 ds1 . b(s2 )f (s2 , t)ds2 ds1 − s = 0
0
0
˜ 0 in L2 (Ω), and that K0 and This gives that −W bD22 |D(A) has a bounded extension K ˜ 0 are given by K ˜ 0 f )(s, t) = s (K
Z
0
1 Z s1
b(s2 )f (s2 , t) ds2 ds1 −
Z
0
0
s Z s1
b(s2 )f (s2 , t) ds2 ds1 .
0
˜ 0 f for f ∈ V . for f ∈ L2 (Ω). Also K0 f = K
Theorem 6.4 The operator K is not compact.
Proof: We will show that there is a bounded sequence (un ) in V , un ∈ D(A), such that W bD22 un does not have a convergent subsequence. Let s0 ∈ (0, 1) and g ∈ C0∞ (0, 1) such that b(s0 ) 6= 0 and g(s0 ) 6= 0. Observe that the function Z 1 Z s1 Z s g1 (s) = b(s2 )g(s2 ) ds2 ds1 − b(σ)g(σ) dσ 0
0
0
is not constant and particularly not identically zero, so that kg1 k > 0. Choose a sequence (In )∞ 1 of pairwise disjoint nonempty open subintervals In of (0, 1), whose lengths we denote by α2n , αn > 0. Let hn ∈ C0∞ (In ) with kh′n k = 1. Observe that Z t h′n (τ ) dτ hn (t) = 0
implies |hn (t)| ≤ αn kh′n k, and thus khn k ≤ αn kh′n k → 0 as n → ∞.
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Then un := g ⊗ hn ∈ C0∞ (Ω) ⊂ D(A), and kun k2V = hun , un i = B(un , un ) = (Aun , un ) and hence kun k2V
Z
1
Z
′′
1
Z
h′′n (t)h(t) dt
1
2
a(s)|g(s)| ds + g (s)g(s)ds 0 0 0 Z 1 Z 1 c(t)|hn (t)|2 dt, a(s)|g(s)|2 ds = kg′ k2 kh′n k2 +
=
Z
1
c(t)|hn (t)|2 dt
0
0
0
which shows that (un ) is bounded in V . Also note that kun k = kgkkhn k → 0 as n → ∞. Putting un,m = un − um , we obtain kK0 un,m k2V
= hK0 un,m , K0 un,m i = B(K0 un,m , K0 un,m ) = (D1 D2 K0 un,m , D1 D2 K0 un,m ) + (acK0 un,m , K0 un,m ).
Note that |(acK0 un,m , K0 un,m )| ≤ ≤
1
˜ 0 un,m k2 sup |a(s)||c(t)|kK
s,t=0 1
˜ 0 k2 kun,m k2 → 0 as m, n → ∞. sup |a(s)||c(t)|kK
s,t=0
From D1 D2 K0 un,m = g1 ⊗ (h′n − h′m ) it therefore follows for n 6= m that kD1 D2 K0 un,m k2 = kg1 k2 kh′n − h′m k2 = kg1 k2 (kh′n k2 + kh′m k2 ), where the last identity is due to h′n being orthogonal to h′m , as the intersection of their supports is empty. Thus lim inf kK0 un,m k2V = 2kg1 k2 > 0,
n>m→∞
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and (K0 un ) cannot have a convergent subsequence in V . Hence K0 is not compact. For b = 0 we thus have σess (K) = {0}, and the span of the eigenvectors of A corresponding to nonzero eigenvalues of K is dense in the orthogonal complement of the kernel of K. In case b 6= 0, however, there are non-zero points in the essential spectrum of K, and a further investigation of the essential spectrum would be necessary to study the completeness of the eigenvectors. In particular, the questions of the location of the essential spectrum, of embedded eigenvalues and of non-zero limit points of eigenvalues arise. Even if the essential spectrum is known, one still needs to find the corresponding range of the spectral projection in order to determine the closure of the span of the eigenvectors corresponding to eigenvalues which are not embedded in the essential spectrum. Allowing for embedded eigenvalues, this set of vectors would be larger but also even harder to describe. Returning to the original two-parameter system, every eigenvalue pair (λ, µ) leads to an eigenvalue λ of K. But the question of whether every eigenfunction of K is the sum of eigenfunctions of the two-parameter system remains, as does that of identifying embedded eigenvalues. This more involved problem may require different techniques (see the discussion in the introduction) and will therefore be considered elsewhere.
References [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, 1965. [2] F.V. Atkinson, Multiparameter Eigenvalue Problems, Vol. 1, Academic Press, 1972. [3] P.A. Binding, P.J. Browne, Third International Workshop on Multiparameter Spectral Theory: abstracts, Editors P.A. Binding and P.J. Browne, University of
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Calgary, 1983. [4] M. Faierman, Two-parameter Eigenvalue Problems in Ordinary Differential Equations, Longman, 1991. [5] M. Faierman, R. Mennicken, A non-standard multiparameter eigenvalue problem in ordinary differential equations, Math. Nachr., 278 (2005), 1550-1560. [6] D. Hilbert, Grundz¨ uge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea, 1953. ¨ rmander, Linear partial differential operators, Springer Verlag, 1976. [7] L. Ho ¨ rmander, The analysis of linear partial differential operators, I, Springer [8] L. Ho Verlag, 1983. [9] T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer Verlag, 1976. [10] A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs 71, American Mathematical Society, 1988. [11] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, 1973. [12] H. Volkmer, Multiparameter Eigenvalue Problems and Expansion Theorems, Lecture Notes in Mathematics 1356, Springer Verlag, 1988.
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