Open Access

Study of the asymptotic eigenvalue distribution and trace formula of a second order operator-differential equation

Boundary Value Problems20112011:7

DOI: 10.1186/1687-2770-2011-7

Received: 2 December 2010

Accepted: 13 July 2011

Published: 13 July 2011

Abstract

The purpose of writing this article is to show some spectral properties of the Bessel operator equation, with spectral parameter-dependent boundary condition. This problem arises upon separation of variables in heat or wave equations, when one of the boundary conditions contains partial derivative with respect to time. To illustrate the problem and the proof in detail, as a first step, the corresponding operator's discreteness of the spectrum is proved. Then, the nature of the eigenvalue distribution is established. Finally, based on these results, a regularized trace formula for the eigenvalues is obtained.

MSC: 34B05; 34G20; 34L20; 34L05; 47A05; 47A10.

Keywords

Hilbert space discrete spectrum regularized trace

Introduction

Let L2 = L2 (H, [0, 1]) H, where H is a separable Hilbert space with a scalar product (·, ·) and a norm ||·|| inside of it. By definition, a scalar product in L2 is
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ1_HTML.gif
(1)

where Y = {y (t), y1}, Z = {z (t), z1} and y(t), z(t) L2 (H, [0, 1]) for which L2 (H, [0, 1]) is a space of vector functions y(t) such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq1_HTML.gif .

Now, consider the equation:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ2_HTML.gif
(2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ3_HTML.gif
(3)
in L2 (H, [0, 1]), where A is a self-adjoint positive-definite operator in H which has a compact inverse operator. Further, suppose the operator-valued function q(t) is weakly measurable, and ||q(t)|| is bounded on [0, 1] with the following properties:
  1. 1.

    q(t) has a second-order weak derivative on [0, 1], and q(l)(t) (l = 0, 1, 2) are self-adjoint operators in H for each t [0, 1], [q(l)(t)]* = q(l)(t), q(l)(t) σ1(H). Here σ1(H) is a trace class, i.e., a class of compact operators in separable Hilbert space H, whose singular values form a convergent series (denoting the compact operator by B, then its singular values are the eigenvalues of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq2_HTML.gif ). If {φ n } is a basis formed by the orthonormal eigenvectors of B, then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq3_HTML.gif . For simplicity, denote the norm in σ1(H) by ||·||1.

     
  2. 2.

    The functions ||q(l)(t)||1 (l = 0, 1, 2) are bounded on [0, 1].

     
  3. 3.

    The relation https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq4_HTML.gif is true for each f H.

     

State that if q(t) ≡ 0, a self-adjoint operator denoted by L0 can be associated with problem (2), (3) whose definition will be given later.

If q(t) 0, the operators L and Q are defined by L = L0 + Q, and Q : Q {y (t), y1} = {q(t) y(t), 0} which is a bounded self-adjoint operator in L2.

After the above definitions and the assumptions, the asymptotic of the eigenvalue distribution and regularized trace of the considered problem will be studied. It is clear that because of the appearance of an eigenvalue parameter in the boundary condition at the end point, the operator associated with problem (2), (3) in L2 (H, [0, 1]) is not self-adjoint. Introduce a new Hilbert space L2 (H, [0, 1]) H with the scalar product defined by formula (1) similar to one used in [1]. Then, in this space, the operator becomes self-adjoint.

In [2], Walter considers a scalar Sturm-Liouville problem with an eigenvalue parameter λ in the boundary conditions. He shows that one can associate a self-adjoint operator with that by finding a suitable Hilbert space. Further, he obtains the expansion theorem by reference to the self-adjointness of that operator. His approach was used by Fulton in [3] later on.

As for the differential operator equations, to the best of this author's knowledge in the articles [1, 46], an eigenvalue parameter appears in the boundary conditions. In [4], the following problem is considered:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equa_HTML.gif

where A = A* > E, and u(x) L2 (H, (0, b)). It is proved that the operator associated with this problem has a discrete spectrum, iff : A has a discrete spectrum. The eigenvalues of this problem form two sequences like https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq6_HTML.gif where n, k N, and μ k is an eigenvalue of A. This is obtained from appearance of λ in the boundary condition.

In [5], both boundary conditions depend on λ. It is shown that the operator defined in the space L2 (H, (0, 1)) H H is symmetric positive-definite. Further, the asymptotic formulas for eigenvalues are obtained.

In this author's previous study [6], for the operator considered in [4], the trace formula has been established.

If h = 0 in (3), then the boundary condition takes the form y(1) = 0. This problem is considered in [[7], Theorem 2.2], where the trace formula is established. It is proved that there exists a subsequence of natural numbers {n m } such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq7_HTML.gif , where μ n and λ n are the eigenvalues of perturbed and non-perturbed operators. For definition of {n m }, see also [[8], Lemma 1].

For a scalar case, please refer to [9], where the following problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equb_HTML.gif

is considered on the interval [0, π]. Then, the sum https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq8_HTML.gif is calculated.

In comparison with the above mentioned articles, here we consider a differential operator equation which has a singularity at 0, and the boundary condition at 1 involves both the eigenvalue parameter λ and physical parameter h < 0.

Problems with λ-dependent boundary conditions arise upon separation of variables in the heat and wave equations. We can also refer to [1017], where boundary-value problems for ordinary differential operators with eigenvalue-dependent boundary conditions are studied.

In 1953, Gelfand and Levitan [18] considered the Sturm-Liouville operator
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equc_HTML.gif

and derived the formula https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq9_HTML.gif , where μ n are the eigenvalues of the above operator. For q (x) ≡ 0 the eigenvalues of the operator are given by λ n = n2.

It is worthwhile to note that, several studies are devoted to searching a regularized trace for the concrete operators (e.g., [918]), as well as differential-operator equations (e.g., [68, 19]) and discrete abstract operators (e.g., [2022]). For further detailed discussion of the subject, please refer to [23].

Trace formulas are used for the approximation of the first eigenvalues of the operators [24, 25] to solve inverse problems [26, 27]. They are also applied to index theory of linear operators [28, 29].

To summarize this study, in Section 1, it is proved that the operator associated with (2), (3) is self-adjoint and has a discrete spectrum. In Section 2, we establish an asymptotic formula for the eigenvalues. To do this, the zeros of the characteristic equation (Lemmas 2.1, 2.2, 2.3) are searched in detail. In Section 3, by using the asymptotic for the eigenvalues, we prove that the series called "a regularized trace" converges absolutely (Lemma 3.1). This enables us to arrange the terms of the series in a suitable way for calculation as in (3.9). To calculate the sum of this series, we introduce a function whose poles are zeros of the characteristic equation, the residues at poles of which are the terms of our series. Finally, we establish a trace formula by integrating this function along the expanded contours.

In conclusion, we apply the results of our study to a boundary value problem generated by a partial differential equation.

1 Definition of L0and proof of discreteness of the spectrum

Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq10_HTML.gif , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq11_HTML.gif is a set of vector functions with values in H (see [30], p. 57) that vanish in the vicinity of zero and are infinitely differentiable in the norm of H. Also, on https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq12_HTML.gif define the operator https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq13_HTML.gif :
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equd_HTML.gif
Using integration by parts it is easy to see that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq13_HTML.gif is symmetric. Denote its closure by L0 and show that it is self-adjoint. To do that, consider the adjoint operator of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq13_HTML.gif as https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq14_HTML.gif . By definition, vector https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq15_HTML.gif if for each https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq16_HTML.gif it holds
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ4_HTML.gif
(1.1)

and Z* = {z* (t), z*} L2. However, using integration by parts from (1.1), it is obvious that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq17_HTML.gif with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq18_HTML.gif and l[z] L2 (H, [0, 1])}. In other words, z(t) has a first-order derivative on [0, 1] which is absolutely continuous in the norm of H and z (0) = z'(0) = 0, Az(t) L2 (H, [0, 1]) and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq19_HTML.gif .

Now, the vector https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq20_HTML.gif if and only if for any https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq21_HTML.gif (1.1) holds, Z* L2 and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq22_HTML.gif .

By virtue of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq24_HTML.gif , we can state that any vector Z from https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq25_HTML.gif must also belong to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq27_HTML.gif . On the other hand, it could be verified that relation (1.1) is also true for
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Eque_HTML.gif

Therefore, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq28_HTML.gif . In other words, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq14_HTML.gif is a self-adjoint operator. However, we know that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq29_HTML.gif . Thus, the closure of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq13_HTML.gif is a self-adjoint operator https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq14_HTML.gif , which we will denote by L0.

By virtue of all as stated above, L0 is defined as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equf_HTML.gif
By the properties of ν ≥ 1, A > E, it follows that L0 is a positive-definite operator. To show that, for each Y D (L0), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equg_HTML.gif

Since the embedding https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq30_HTML.gif is continuous ([[31], Theorem 1.7.7], [[32], p. 48]), then, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq31_HTML.gif , where c > 0 is a constant.

Thus,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equh_HTML.gif

which shows that L0 is a positive-definite operator.

To prove the discreteness of the spectrum, we will use the following Rellich's theorem (see [[33], p. 386]).

Theorem 1.1. Let B be a self-adjoint operator in H satisfying (, φ) ≥ (φ, φ), φ D B , where D B is a domain of B.

Then, the spectrum of B is discrete if and only if the set of all vectors φ D B , satisfying (, φ) ≤ 1 is precompact.

Let γ1γ2 ≤ · · · ≤ γ n ≤ · · · be the eigenvalues of A counted with multiplicity and φ1, φ2,..., φ n ,... be the corresponding orthonormal eigenvectors in H.

Take y k (t) = (y (t), φ k ). Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ5_HTML.gif
(1.2)

Hence, using the Rellich's theorem, we come to the following theorem:

Theorem 1.2. If the operator A-1 is compact in H, then the operator L0 has a discrete spectrum.

Proof. By virtue of positive-definiteness of L0, by Rellich's theorem, it is sufficient to show that the set of vectors
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ6_HTML.gif
(1.3)

is precompact in L2.

To prove this theorem, consider the following lemma.

Lemma 1.1. For any given ε > 0, there is a number R = R(ε), such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equi_HTML.gif
Proof. From (1.1) for Y Y :
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equj_HTML.gif
Since γ R → ∞ for R → ∞, for any given ε > 0, we could choose R(ε) such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq32_HTML.gif . Therefore, for this choice of R the inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ7_HTML.gif
(1.4)
holds. On the other hand, by virtue of (1.3):
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equk_HTML.gif
From (1.4) and the above, it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equl_HTML.gif

This proves Lemma 1.1.

Now, turn to the proof of Theorem 1.2. Assume, Y Y. Denote the set of all vector-functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq33_HTML.gif , by E R . Then, from Lemma 1.1 it follows that for the set Y, E R is an ε-net in L2. Therefore, to prove the precompactness of the set Y, we must prove the precompactness of E R in L2. Since |y k (1)| ≤ 1 (k = 1,..., R), it is sufficient to show that y k (t) (k = 1,..., R) satisfies the criteria of precompactness in L2 (0, 1) [[34], p. 291]. In other words, y k (t), (k = 1,..., R) must be equicontinuous and bounded with respect to the norm in L2 (0, 1). To show that, using (1.3) results in
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equm_HTML.gif
which proves the boundedness of the functions y k (t) (k = 1,..., R). Assume that y k (t) is a zero outside the interval (0, 1). Then, by using the following relation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equn_HTML.gif
we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ8_HTML.gif
(1.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ9_HTML.gif
(1.6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ10_HTML.gif
(1.7)
From the above, for |η| < ε we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equo_HTML.gif

This shows the equicontinuity of E R , and it completes the proof of the discreteness of the spectrum of L0.

2 The derivation of the asymptotic formula for eigenvalue distribution of L0

Suppose that the eigenvalues of A are γ n ~ an α (n → ∞, a > 0, α > 0). Then, by virtue of the spectral expansion of the self-adjoint operator A, we get the following boundary value problem for the coefficients y k (t) = (y(t), φ k ):
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ11_HTML.gif
(2.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ12_HTML.gif
(2.2)
The solution to problem (2.1) from L2 (0, 1) is
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equp_HTML.gif
For this solution to satisfy (2.2), it is necessary and sufficient to hold
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ13_HTML.gif
(2.3)
at least for one γ k (λγ k ). Therefore, the spectrum of the operator L0 consists of those real values of λγ k , such that at least for one k
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ14_HTML.gif
(2.4)
where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq34_HTML.gif . Then, by using (2.4) and identity https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq35_HTML.gif [[35], p. 56], we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ15_HTML.gif
(2.5)
Find the eigenvalues of the operator L0 which are less than γ k . These values correspond to the imaginary roots of Equation 2.5. By taking https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq36_HTML.gif and using [[35], p. 51]:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equq_HTML.gif
we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equr_HTML.gif
or equivalently
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ16_HTML.gif
(2.6)
Now, consider the quadratic equation https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq37_HTML.gif whose roots are given as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equs_HTML.gif
Therefore, the coefficients for y n in (2.6) become positive for
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ17_HTML.gif
(2.7)

Further, let N be the number of positive roots of the function in (2.6), and W be the number of sign changes in its coefficients. Because the radius of convergence of this series is ∞, then by Descartes' rule of signs [[36], p. 52] W - N is a nonnegative even number. From (2.7), W = 1, therefore N = 1. Hence, beginning with some k, Equation 2.6 has exactly one positive root corresponding to the imaginary root of Equation 2.5.

Now, find the asymptotic of the imaginary roots of Equation 2.5. For z = iy and using the asymptotic of J ν (z) for imaginary z a large |z| [[37], p. 976]
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equt_HTML.gif
This means (2.4) is equivalent to
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equu_HTML.gif
from which
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ18_HTML.gif
(2.8)
Using (2.8) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq38_HTML.gif , we come up with the asymptotic formula for the eigenvalues of L0 which are less than γ k
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ19_HTML.gif
(2.9)
Now, find the asymptotic of those solutions of Equation 2.3 which are greater than γ k , i.e., the real roots of Equation 2.5. By virtue of the asymptotic for a large |z| [[35], p. 222]
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equv_HTML.gif
Equation 2.5 becomes
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equw_HTML.gif
Hence,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ20_HTML.gif
(2.10)

where m is a large integer. Therefore, we can state the following Lemma 2.1:

Lemma 2.1. The eigenvalues of the operator L0 form two sequences
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equx_HTML.gif

where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq39_HTML.gif . Denote the imaginary and real roots of Equation 2.2 by x0,kand xm, k, respectively.

State the following two lemmas.

Lemma 2.2. Equation 2.5 has no complex roots except the pure imaginary or real roots.

Proof. λ is real since it is eigenvalue of self-adjoint operator associated with problem (2.1), (2.2). γ k is real by our assumption (A* = A). Hence, the roots of (2.5) are square roots of real numbers. Lemma 2.2 is proved.

Let C be a rectangular contour with vertices at ±iB, ±iB + A m , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq40_HTML.gif , and B is a large positive number. Further, assume that this contour bypasses the origin and the imaginary root at -ix0,kalong the small semicircle on the right side of the imaginary axis and ix0,kon the left.

Then, we claim that the following lemma is true.

Lemma 2.3. For a sufficiently large integer m, the number of zeros of the function
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equy_HTML.gif

inside of C is exactly m.

Proof. Since https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq41_HTML.gif is an entire function of z, then the number of its zeros inside of C equals:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equz_HTML.gif
In the above, we have used the following identities:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equaa_HTML.gif
As the integrand is an odd function. the order of its numerator in the vicinity of zero is O(zν+1), and the order of its denominator is O(z ν ), the integral along the left part of contour vanishes. Now, consider the integrals along the remaining three sides of the contour. On these sides [[35], p. 221, p. 88]
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equab_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equac_HTML.gif

η1,ν (z) and η2,ν (z) are of order https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq42_HTML.gif for large |z|.

For simplicity, denote the integrand by f(z), then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equad_HTML.gif

One can analogously show that the integral along the lower side tends to the same number.

To calculate the integral along the fourth side, use the relations: https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq43_HTML.gif for large |z|, and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq44_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq42_HTML.gif is bounded on the right-hand side of the contour, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equae_HTML.gif

Consequently, the limit of the integral along the entire contour is https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq45_HTML.gif . However, as the integral must be an integer, it should be equal to m. This completes the proof of Lemma 2.3.

By using the above results, derive the asymptotic formula for the eigenvalue distribution of L0. To do that, denote the eigenvalue distribution of the operator L0 by N (λ, L0). Then:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equaf_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equag_HTML.gif
Since γ k ~ a · k α , then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq46_HTML.gif . That is
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ21_HTML.gif
(2.11)
From Lemmas 2.2 and 2.3 and the asymptotic of xm, k, it follows that one can find a number c such that for a large m
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equah_HTML.gif
From this inequality, it follows that N2(λ) is less than https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq47_HTML.gif , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq47_HTML.gif is the number of the positive integer pairs (m, k) satisfying the inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ22_HTML.gif
(2.12)
Also, N2(λ) is greater than https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq48_HTML.gif , where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq48_HTML.gif is the number of the positive integer pairs for which
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ23_HTML.gif
(2.13)
To summarize, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ24_HTML.gif
(2.14)
Thus, by (2.12) and (2.13) as in [[38], Section 3, Lemma 2] we have:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equai_HTML.gif

where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq49_HTML.gif .

From the above, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ25_HTML.gif
(2.15)
Therefore, by virtue of (2.11) and (2.15), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equaj_HTML.gif
For α > 2
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equak_HTML.gif

and consequently, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq51_HTML.gif .

For α > 2, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq52_HTML.gif or, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq53_HTML.gif .

For α = 2, N (λ) ~ (c1 + c2) λ from which λ n (L0) ~ dn, d = (c1 + c2)-1.

Then, as Q is a bounded operator in L2, it follows from the relation for the resolvents of the operators L0 and L [[30], p. 219]
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equal_HTML.gif
that the spectrum of L is also discrete. By virtue of the last equality and the properties that hold for s numbers of compact operators [[30], pp. 44, 49] as in [[38], Section 3, Lemma 2], for the eigenvalues of L denoted by μ n (L), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equam_HTML.gif

Therefore, we can state the following theorem:

Theorem 2.1. If γ n ~ an α (0 < a, α > 0), then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ26_HTML.gif
(2.16)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equan_HTML.gif

For simplicity, we will denote the eigenvalues of L0 and L by λ n and μ n , respectively.

3 Regularized trace of the operator L

Now make use of the theorem proved in [20] for abstract operators. At first, introduce the following notations.

Let A0 be a self-adjoint positive discrete operator, {λ n } be its eigenvalues arranged in ascending order, {φ j } be a basis formed by the eigenvectors of A0, B be a perturbation operator, and {μ n } be the eigenvalues of A0 + B. Also, assume that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq54_HTML.gif . For operators A0 and B in [[20], Theorem 1], the following theorem is proved.

Theorem 3.1. Let the operator B be such that D(A0) D(B), and let there exist a number δ [0, 1) such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq55_HTML.gif has a bounded extension, and number ω [0, 1), ω + δ < 1 such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq56_HTML.gif is a trace class operator. Then, there exists a subsequence of natural numbers https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq57_HTML.gif and a subsequence of contours Γ m C, that for ωδ the formula
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equao_HTML.gif

is true.

Note that the conditions of this theorem are satisfied for L0 and L. That is, if we take A0 = L0, B = Q, then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq58_HTML.gif is bounded. For https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq59_HTML.gif and α > 2, from asymptotic (2.16), we will have that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq60_HTML.gif is a trace class operator. If α < 2, then https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq61_HTML.gif will be a trace class operator for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq62_HTML.gif .

Thus, by the statement of Theorem 3.1, for α > 2, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ27_HTML.gif
(3.1)

where ψ1(x), ψ2(x),... are the orthonormal eigenvectors of L0.

Introduce the following notation:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ28_HTML.gif
(3.2)

and investigate the sum of series https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq63_HTML.gif , which as will be seen later, is independent of the choice of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq57_HTML.gif . We will call the sum of this series a regularized trace of the operator L0.

Now, we calculate the norm for the eigen-vectors of the operator L0 in L2. To do this, we will use the following identity obtained from the Bessel equation"
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equap_HTML.gif
As αβ, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ29_HTML.gif
(3.3)
We also consider the following identities:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equaq_HTML.gif
By the above identities and also by the equation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equar_HTML.gif
satisfied by xm, k, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equas_HTML.gif
Therefore,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equat_HTML.gif
So, the orthonormal eigen-vectors of L0 are
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ30_HTML.gif
(3.4)

Now, we prove the following lemma.

Lemma 3.1. If the operator function q(t) has properties 1, 2, and also α > 0, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ31_HTML.gif
(3.5)

Proof. Assume that f k (t) = (q(t) φ k , φ k ). By Lemma 2.1 we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq64_HTML.gif .

So, in virtue of the inequality https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq65_HTML.gif [[35], p. 666] and properties 1 and 2 we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equau_HTML.gif

To estimate the second series in (3.5), we use the relation https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq66_HTML.gif .

By hypothesis of Lemma 3.1 α > 0. Therefore, denoting this sum by s, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equav_HTML.gif

This proves Lemma 3.1.

Now, assume that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ32_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ33_HTML.gif
(3.7)

for small δ > 0.

Then, we can state the following theorem.

Theorem 3.2. Let the conditions of Theorem 2.1, (3.6) and (3.7) hold. If the operator-value function q(t) has properties 1-3, then the following formula is true
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ34_HTML.gif
(3.8)
Proof. By virtue of lemma 3.1 we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ35_HTML.gif
(3.9)
At first evaluate the inner sum in the second term on the right hand side of (3.9). To do this, as N → ∞ investigate the asymptotic behavior of the function
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equaw_HTML.gif

To derive a formula for R N (t), show for each fixed value of k, the m th term of the sum R N (t) as a residue at the point xm, kof some complex variable function with poles at https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq67_HTML.gif .

For this purpose, consider the following function:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equax_HTML.gif
The poles of this function are x0,k,...,xN - 1,kand j1,..., j N (J ν (j n ) = 0). The residue at j n equals
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equay_HTML.gif
Now, compute the residue at xm, k:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ36_HTML.gif
(3.9a)
Denote the right hand side of (3.10) by G(z). Since xm, ksatisfies equation (2.4), by setting z = xm, kand using the identity
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equaz_HTML.gif
we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equba_HTML.gif
Therefore,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbb_HTML.gif

Consider the contour C mentioned in Lemma 2.3 as the contour of integration. According to Lemmas 2.1 and 2.3, for a sufficiently large N, we have xN - 1,k< A N < xN, kand j N < A N < jN+1.

It could easily be verified that in the vicinity of zero, the function g(z) is of order O (z ν ). By virtue of this asymptotic and because g(z) is an odd function, the integral along the left-hand side of the contour C vanishes when r (radius of a semicircle) goes to zero.

Furthermore, if z = u + iv, then for large |v| and u ≥ 0, the integrand will be of order O (e|v|(2t- 2)). That is, for a given value of A N , the integrals along the upper and lower sides of C go to zero as B → ∞ (0 < t < 1). Thus, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ37_HTML.gif
(3.10)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbc_HTML.gif
Also, along the contour C for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq68_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq69_HTML.gif , we have |tz| → ∞. Therefore, in integral (3.11), we could replace the Bessel functions by their asymptotic at large arguments. Hence, from
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbd_HTML.gif
as N → ∞ we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ38_HTML.gif
(3.11)
Denote the right side of (3.12) by J:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ39_HTML.gif
(3.12)
Then the limit of (3.11) becomes:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ40_HTML.gif
(3.13)
Using (3.6) and (3.13), we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ41_HTML.gif
(3.14)
Moreover, if (3.7) holds, then by virtue of the known relation for a large N [[35], p. 642]
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Eqube_HTML.gif
Hence, we will have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ42_HTML.gif
(3.15)
Using property 2 and the asymptotic of xm, k
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ43_HTML.gif
(3.16)
Earlier it was obtained that under the assumptions 1-3 (see [[7], Theorem 2.2])
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ44_HTML.gif
(3.17)
Thus, from (3.14) to (3.18), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbf_HTML.gif
Consequently,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ45_HTML.gif
(3.18)
In a similar way to the one considered above, we get (this time Equation 2.5 has no imaginary roots, so the contour C will only bypass the origin on the right half-plane):
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ46_HTML.gif
(3.19)
Finally, combining (3.19) and (3.20), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbg_HTML.gif

which completes the proof.

Remark. It should be noted that in condition 1, property q(l)(t) σ1, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq70_HTML.gif may be weakened. Namely, we may just require to hold
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbh_HTML.gif

Then formula (3.8) takes the form https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq71_HTML.gif . There exist the bounded functions that are not from the trace class, even compact, but satisfy the above stated condition. Now, introduce an example.

Example. We consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ47_HTML.gif
(3.20)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ48_HTML.gif
(3.21)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ49_HTML.gif
(3.22)
in the cylinder ∂Ω × [0, 1], where Ω is a circle in R2 ((x, y) R2) of radius 1. Also, ∂Ω is a circumference of this circle, n is an exterior normal to the surface ∂Ω × [0, 1] and h = const.. Looking for the solution of this problem, which can be represented as u(x, y, z, t) = U(x, y, z)T(t), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbi_HTML.gif
Thus, the left-hand side of this equality depends only on t, while the right-hand side on x, y, z. This means they are equal to some constant which we will denote by -λ. Therefore,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbj_HTML.gif
and (3.22) becomes like
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ50_HTML.gif
(3.23)
Using the cylindric coordinates x = r cos φ, y = r sin φ, z = z, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbk_HTML.gif
The boundary condition in (3.24) becomes
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ51_HTML.gif
(3.24)
We will solve this problem by separation of variables. Taking U(r, φ, z) = V (r, z)ϕ(φ), q(r, φ, z) = Q(r, z), and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq72_HTML.gif , ν = const., we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbl_HTML.gif
By making https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq73_HTML.gif substitution, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ52_HTML.gif
(3.25)
and (3.25), (3.23) take the form:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ53_HTML.gif
(3.26)
where Q(r, z) is a real-valued function which is continuous on [0, 1] × [0, 1], and has second partial derivative with respect to r on [0, 1] for each fixed z. Fourier series of this function and its partial derivatives converge, respectively, to their values. Also assume that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbm_HTML.gif
Now, rewrite the problem in the differential operator form:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ54_HTML.gif
(3.27)
where v(r) = V1(r, ·) is a vector function with the values from L2(0, 1). Operators A and q(r) are defined in the following way:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equ55_HTML.gif
(3.28)
Obviously, the operator A is self-adjoint, positive-definite, and A-1 is a compact operator in L2(0, 1). Also, the eigenvalues of A are of the form:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbn_HTML.gif

Then, by virtue of Theorem 2.1, the eigenvalues of this problem behave like https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq74_HTML.gif .

Using the statement of Theorem 3.2, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbo_HTML.gif
where μ i are the eigenvalues of problem (3.28) with q(r) ≡ 0. Now calculate https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_IEq75_HTML.gif
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbp_HTML.gif
In a similar way, we can find
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbq_HTML.gif
Therefore,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-7/MediaObjects/13661_2010_Article_7_Equbr_HTML.gif

The authors declare that they have no competing interests

Declarations

Acknowledgements

The author would like to express his thanks to Dr. Yaghoob Ebrahimi, U.S.Fulbright Scholar assigned to Khazar University during 2009-10 academic year, for the latter's help in editing, interpretation, and modification of the initial version of this study.

Authors’ Affiliations

(1)
Department of Differential Equation, Institute of Mathematics and Mechanics-Azerbaijan National Academy of Science
(2)
Mathematics Department, Khazar University

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© Aslanova; licensee Springer. 2011

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