Computation of eigenvalues and eigenfunctions of a Schrödinger‐type boundary‐value‐transmission problem with retarded argument

Abstract

MSC Classification: 34L20; 35R10The purpose of this study is to investigate a Schrödinger type of boundary-value-transmission problem with retarded argument. By modifying some techniques of [S.B. Norkin, Differential equations of the second order with retarded argument, Translations of Mathematical Monographs, Vol. 31, AMS, Providence, RI, 1972] and suggesting own approaches, we find asymptotic formulas for the eigenvalues and eigenfunctions of the second-order differential equation with retarded argument, which includes the eigenparameter as a quadratic function. KEYWORDS asymptotics of eigenvalues and eigenfunctions, differential equation with retarded argument, transmission conditionsHere, q(x) is a real valued function and continuous in I with a finite limit q (b ± 0) = lim x→b±0 q(x); the real valued function Δ(x) ⩾ 0 continuous in I and has a finite limit c]; the real valued function p(x) is continuous in I such that p(b) = p(c) = p 0 (p 0 ∈ R); is a real eigenparameter; and , , ≠ 0 are arbitrary real numbers. 6604

Oscillation Properties for Non-Classical Sturm-Liouville Problems With Additional Transmission Conditions

“…In this work we will extend and generalize some well-known Sturm's comparison and oscillation theorems for problems of a new type, which consist of Sturm-Liouville equation In recent years, the authors of this study and some other mathematicians have studied many other spectral properties of similar BTVP's (see, for example, [1,2,4,5,6,7,9,13,17,22,23,27,28]). BVTP's also find many important application in the study of various phenomena in physics and technology, such as vibration of string involving different types of loads (see, for example, [12,14,18,21,25,26]), heat transfer through a solid-liquid interface(see, for example, [8,15,20]), water vapor diffusion through a porous membrane (see, for example, [24]).…”

Two‐linked periodic Sturm–Liouville problems with transmission conditions

“…In a previous study, 13 a new technique is developed to estimate the eigenvalues of non-selfadjoint Sturm-Liouville problems with periodic and anti-periodic boundary conditions. In recent years, there has been growing interest of Sturm-Liouville problems with additional transmission conditions at some interior points of interaction, because of the appearance of new and interesting applications in mathematical physics (see, e.g., previous studies [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] and the references therein). For example, in electrostatics and magnetostatics, the model problem which describes the heat transfer through an infinitely conductive layer is a transmission problem.…”

“…In recent years, there has been growing interest of Sturm–Liouville problems with additional transmission conditions at some interior points of interaction, because of the appearance of new and interesting applications in mathematical physics (see, e.g., previous studies 14–28 and the references therein). For example, in electrostatics and magnetostatics, the model problem which describes the heat transfer through an infinitely conductive layer is a transmission problem 29 .…”

Non-classical periodic boundary value problems with impulsive conditions

“…Recently, there has been an increasing interest in Sturm-Liouville boundary value problems defined on two or more disjoint segments with common ends, the so-called many-interval SLPs (see, for example, [9][10][11][12][13][14][15][16][17][18][19][20][21] and references cited therein). To deal with such multi-interval boundary value problems, naturally, additional conditions (the so-called transmission conditions, jump conditions, interface conditions, and impulsive conditions) are imposed at these common endpoints.…”