Spectral analysis of wave propagation on branching strings

Rakib F Efendiev,Hamzaga D Orudzhev,Zaki Fa El-Raheem

doi:10.1186/s13661-016-0723-3

Rakib F Efendiev, Hamzaga D Orudzhev + Show 1 more

Open Access

https://doi.org/10.1186/s13661-016-0723-3

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Abstract

In this paper the classical Hill problem with complex potentials are extended to the star graph. The definition of the Hill operator on such graph is discussed. The operator is defined with complex, periodic potentials and using special boundary conditions connecting values of the functions at the vertices. An explicit description of the resolvent is given and the spectrum is described exactly, the inverse problem with respect to the reflection coefficients is solved.

Highlights

1 Introduction The purpose of the present paper is the spectral analysis of a wave propagation in a layered, inhomogeneous medium, such as a branching tube or a system of joined strings
It is well known [ – ] that wave propagation in a one-dimensional non-conservative medium in a frequency domain is described by the Schrödinger equation
As a model of layered, inhomogeneous medium we will use a special type of noncompact graph, star graph, that is, a flexible mathematical construction with single vertex in which a finite number of edges Nk = [, ∞), k =, . . . , n, are joined

Summary

Introduction

On each edge the wave function is a solution of the one-dimensional equation. Note that every solution k(xk, λ) of the problem on the edge Nk = [ , ∞), k = , , , is a linear combination of the functions ykk(xk, λ), yjk(xk, λ), j = k, j, k = , , , and can be written in the form k(xk, λ) = C (kj)(xk)ykk(xk, λ) + C (kj )(xk)yjk(xk, λ), j = k, j, k = , , , where C (kj)(xk) and C (kj )(xk) are such that conditions ( )-( ) hold for We will construct the resolvent of the operator LG for Im λ > .

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Conclusion