Spectral properties of generalized eigenparameter dependent discrete Sturm-Liouville type equation (original) (raw)

Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications

Opuscula Mathematica

Let \(n\in\mathbb{N}^{*}\), and \(N\geq n\) be an integer. We study the spectrum of discrete linear \(2n\)-th order eigenvalue problems \[\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}\] where \(\lambda\) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear \(2n\)-th order problems by applying the variational methods and critical point theory.

Difference equations of second order with spectral singularities

Journal of Mathematical Analysis and Applications, 2003

In this paper using the uniqueness theorem of analytic functions we investigated the eigenvalues and the spectral singularities of the difference equation a n−1 y n−1 + b n y n + a n y n+1 = λy n , n∈ Z = {0, ±1, ±2, . . .}, where t{a n } n∈Z , {b n } n∈Z are complex sequences and λ is a spectral parameter.

On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition

Hacettepe Journal of Mathematics and Statistics, 2019

The spectral problem\[-y''+q(x)y=\lambda y,\ \ \ \ 0<x<1\]\[y(0)=0, \quad y'(0)=\lambda(ay(1)+by'(1)),\]is considered, where lambda\lambdalambda is a spectral parameter, q(x)inL1(0,1)q(x)\in{{L}_{1}}(0,1)q(x)inL1(0,1) is a complex-valued function, aaa and bbb are arbitrary complex numbers which satisfy the condition ∣a∣+∣b∣ne0|a|+|b|\ne 0a+bne0. We study the spectral properties (existence of eigenvalues, asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions in Lp(0,1){{L}_{p}}(0,1)Lp(0,1)) of the above-mentioned Sturm-Liouville problem.

Characterization of the Spectrum of an Irregular Boundary Value Problem for the Sturm-Liouville Operator

Boundary Value Problems, Integral Equations and Related Problems, 2010

We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L 2 (0, π) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator. In the present paper, we consider the eigenvalue problem for the Sturm-Liouvulle equation u ′′ − q(x)u + λu = 0 (1) on the interval (0, π) with the boundary conditions u ′ (0) + (−1) θ u ′ (π) + bu(π) = 0, u(0) + (−1) θ+1 u(π) = 0, (2) where b is a complex number, θ = 0, 1, and the function q(x) is an arbitrary complex-valued function of the class L 2 (0, π). Denote by c(x, µ), s(x, µ) (λ = µ 2) the fundamental system of solutions to (1) with the initial conditions c(0, µ) = s ′ (0, µ) = 1, c ′ (0, µ) = s(0, µ) = 0. The following identity is well known c(x, µ)s ′ (x, µ) − c ′ (x, µ)s(x, µ) = 1. (3) Simple calculations show that the characteristic equation of (1), (2) can be reduced to the form ∆(µ) = 0, where

On the Neumann eigenvalues for second-order Sturm–Liouville difference equations

Advances in Difference Equations, 2020

The paper is concerned with the Neumann eigenvalues for second-order Sturm–Liouville difference equations. By analyzing the new discriminant function, we show the interlacing properties between the periodic, antiperiodic, and Neumann eigenvalues. Moreover, when the potential sequence is symmetric and symmetric monotonic, we show the order relation between the first Dirichlet eigenvalue and the second Neumann eigenvalue, and prove that the minimum of the first Neumann eigenvalue gap is attained at the constant potential sequence.

A class of sturm-liouville operators with eigenparameter dependent boundary and transmission conditions

International Journal of Pure and Applied Math. Vol. 84 (2013), pp. 149-157

""In this study, we investigate a Sturm-Liouville type problem with eigenparameter dependent boundary conditions and eigenparameter dependent transmission conditions. By establishing a new self-adjoint operator A associated with the problem, we construct fundamental solutions and obtain asymptotic formulae for its eigenvalues and fundamental solutions. ""

The Finite Spectrum of Sturm-Liouville Operator With δ-Interactions 1

The goal of this paper is to study the finite spectrum of Sturm-Liouville operator with δinteractions. Such an equation gives us a Sturm-Liouville boundary value problem which has n transmission conditions. We show that for any positive numbers m j (j = 0, 1, ..., n) that are related to number of partition of the intervals between two successive interaction points, we can construct a Sturm-Liouville equations with δ-interactions, which have exactly d eigenvalues. Where d is the sum of m j 's.

On the spectrum of an irregular Sturm-Liouville problem

Doklady Mathematics, 2010

We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L 2 (0, π) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.

On two-point boundary value problems for the Sturm-Liouville operator

arXiv: Spectral Theory, 2015

In this paper, we study spectral problems for the Sturm-Liouville operator with arbitrary complexvalued potential q(x) and two-point boundary conditions. All types of mentioned boundary conditions are considered. We ivestigate in detail the completeness property and the basis property of the root function system.

Nonlinear discrete Sturm–Liouville problems with global boundary conditions

Journal of Difference Equations and Applications, 2012

This paper is devoted to the study of nonlinear difference equations subject to global nonlinear boundary conditions. We provide sufficient conditions for the existence of solutions based on properties of the nonlinearities and the eigenvalues of an associated linear Sturm-Liouville problem.