The finite spectrum of Sturm–Liouville problems with transmission conditions

doi:10.1016/j.amc.2011.05.033

Applied Mathematics and Computation

Volume 218, Issue 4, 15 October 2011, Pages 1166-1173

https://doi.org/10.1016/j.amc.2011.05.033 Get rights and content

Abstract

We study the finite spectrum of Sturm–Liouville problems with transmission conditions. For any positive integer n, we construct a class of regular Sturm–Liouville problems with transmission conditions, which have exactly n eigenvalues, and these n eigenvalues can be located anywhere in the complex plane in non-self-adjoint case and anywhere along the real line in the self-adjoint case.

Introduction

In recent years the Sturm–Liouville problems with transmission conditions have been an important research topic in mathematical physics [1], [2], [3], [4], [5], [6], [7]. Boundary value problems with discontinuity conditions inside the interval often appear in applications. Such problems are connected with discontinuous material properties, such as heat and mass transfer, varied assortment of physical transfer problems, vibrating string problems when the string loaded additionally with point masses, and diffraction problems [6], [7]. The study of the structure of the solution of the matching region leads to the consideration of an eigenvalue problem for a second order differential operator with piecewise continuous coefficients and transmission conditions at interior points.

As is well-known, the classical Sturm–Liouville theory states that the spectrum of a regular or singular, self-adjoint Sturm–Liouville problem (SLP) is unbounded and therefore infinite. This result is generally established under the assumption that the leading coefficient p and the weight function w are both positive. Atkinson in his book [8] suggested that if the coefficients of SLP satisfy some conditions, the problem may have finite eigenvalues. But he did not elaborate with either an example or a theorem. In 2001, Kong et al. [9] constructed a class of SLPs with exactly n eigenvalues for any positive integer n. Furthermore, Kong et al. [10] gave these kinds of Sturm–Liouville problems with self-adjoint boundary conditions (either separated or coupled) with matrix representations. This further illustrate the reality that the SLPs have finite spectrum. Now, one question arises. What about the SLPs with transmission conditions? Whether they also have finite spectrum? We will show, in this paper, that the answer is definite. We construct a class of SLPs with transmission conditions with exactly n eigenvalues, and show that these n eigenvalues can be located anywhere in the complex plane in the non-self-adjoint case and anywhere on the real line in the self-adjoint case. As in [9] our construction based on the characteristic function whose zeros are the eigenvalues. The key to this analysis is an iterative construction of the characteristic function. At the end of this paper we illustrate our results by an example.

Section snippets

Notation and preliminaries

We consider the SLP consisting of the equation $- ({py}^{'})^{'} + qy = λ wy, on J = (a, c) \cup (c, b), c \in (a, b), with - \infty < a < b < + \infty,$ together with boundary conditions of the form $AY (a) + BY (b) = 0, Y = [\begin{matrix} y \\ {py}^{'} \end{matrix}], A, B \in M_{2} (C)$ and the transmission conditions $CY (c -) + DY (c +) = 0,$ where A = (a_ij)_2×2, B = (b_ij)_2×2 are complex valued 2 × 2 matrices, and C = (c_ij)_2×2, D = (d_ij)_2×2 are real valued 2 × 2 matrices satisfying det(C) = ρ > 0, det(D) = θ > 0. $M_{2} (C)$ denotes the set of square matrices of order 2 over $C$ . Here λ is the spectral parameter, and the coefficients satisfy

The finite spectrum of SLPs with transmission conditions

In this section we assume (2.4) holds and there exists a partition of interval J $a = a_{0} < a_{1} < a_{2} < \dots < a_{2 m} < a_{2 m + 1} = c, c = b_{0} < b_{1} < b_{2} < \dots < b_{2 n} < b_{2 n + 1} = b$ for some integers m and n, such that $\begin{matrix} r = \frac{1}{p} = 0 on (a_{2 k}, a_{2 k + 1}), \int_{a_{2 k}}^{a_{2 k + 1}} w \neq 0, k = 0, 1, \dots, m, \\ r = \frac{1}{p} = 0 on (b_{2 i}, b_{2 i + 1}), \int_{b_{2 i}}^{b_{2 i + 1}} w \neq 0, i = 0, 1, \dots, n; \end{matrix}$ and $\begin{matrix} q = w = 0 on (a_{2 k + 1}, a_{2 k + 2}), \int_{a_{2 k + 1}}^{a_{2 k + 2}} r \neq 0, k = 0, 1, \dots, m - 1, \\ q = w = 0 on (b_{2 i + 1}, b_{2 i + 2}), \int_{b_{2 i + 1}}^{b_{2 i + 2}} r \neq 0, i = 0, 1, \dots, n - 1 . \end{matrix}$ We also need some additional notations. Given (3.1), (3.2), (3.3), let $\begin{matrix} r_{k} = \int_{a_{2 k + 1}}^{a_{2 k + 2}} r, k = 0, 1, \dots, m - 1, q_{k} = \int_{a_{2 k}}^{a_{2 k + 1}} q, w_{k} = \int_{a_{2 k}}^{a_{2 k + 1}} w, k = 0, 1, \dots, m, \\ {\tilde{r}}_{i} = \int_{b_{2 i + 1}}^{b_{2 i +}} \end{matrix}$

Acknowledgments

We express our thanks to Professor Anton Zettl for his suggestion of this research. This work was Supported by National Natural Science Foundation of China (Grant No. 10861008), the “211 Project” Innovative Talents Training Program of Inner Mongolia University and a Grant-in-Aid for Scientific Research from Inner Mongolia University of Technology (Grant No. ZS201032). The authors thank the reviewers valuable comments and suggestions.

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View full text

The finite spectrum of Sturm–Liouville problems with transmission conditions

Abstract

Introduction

Section snippets

Notation and preliminaries

The finite spectrum of SLPs with transmission conditions

Acknowledgments

Appl. Math. Comput.

Acta Math. Scientia

Rep. Math. Phys.

J. Math. Anal. Appl.

J. Differential Equations

J. Differential Equations

An exactly solvable periodic Schroedinger operator

J. Phys. A: Math. Gen.