SPECTRAL PROPERTIES OF A NON SELFADJOINT SYSTEM OF DIFFERENTIAL EQUATIONS WITH A SPECTRAL PARAMETER IN THE BOUNDARY CONDITION

E Kir,C Yanik,G Bascanbaz-Tunca

doi:10.4067/s0716-09172005000100005

E Kir, C Yanik + Show 1 more

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https://doi.org/10.4067/s0716-09172005000100005

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Abstract

In this paper we investigated the spectrum of the operator L(?) generated in Hilbert Space of vector-valued functions L2 (R+, C2) by the system iy0 1 + q1 (x) y2 = ?y1, ?iy0 2 + q2 (x) y1 = ?y2 (0.1) , x ?R+ := [0,?), and the spectral parameter- dependent boundary condition (a1? + b1) y2 (0, ?) ? (a2? + b2) y1 (0, ?)=0, where ? is a complex parameter, qi, i = 1, 2 are complex-valued functions ai 6= 0, bi 6= 0, i = 1, 2 are complex constants. Under the condition sup x?R+ {exp ?x |qi (x)|} < ?, i = 1, 2,?> 0, we proved that L(?) has a finite number of eigenvalues and spectral singularities with finite multiplicities. Furthermore we show that the principal functions corresponding to eigenvalues of L(?) belong to the space L2 (R+, {C2) and the principal functions corresponding to spectral singularities belong to a Hilbert space containing L2 (R+, C2).

Highlights

Introduction(Schwartz [8] named these points as spectral singularities of L). Naimark has proved that spectral singularities are on the continuous spectrum, he has shown that L has a finite number of eigenvalues and spectral singularities with finite multiplicities if the condition
Let us consider the nonself-adjoint one dimensional Schrodinger operator L generated in L2 (R+) by the differential expression l(y) = −y00 + q (x) y, x ∈ R+and the boundary condition y (0) = 0 as Ly = ly, where q is a complexvalued function
In this paper we investigated the spectrum of the operator L(λ) generated in Hilbert Space of vector-valued functions L2 (R+, C2) by the system

Summary

Introduction

(Schwartz [8] named these points as spectral singularities of L). Naimark has proved that spectral singularities are on the continuous spectrum, he has shown that L has a finite number of eigenvalues and spectral singularities with finite multiplicities if the condition. The spectrum of the operator generated by the system (1.1) with the boundary condition y2 (0) − hy (0) = 0, (which is the special case of (1.2) when ai = 0, b = 1) here h 6= 0 is a complex constant, has been investigated in [5] and in [1]. We discussed the spectrum of L (λ) defined by (1.1) and (1.2) and proved that L (λ) has a finite number of eigenvalues and spectral singularities with finite multiplicities under the conditions.

Preliminaries

Eigenvalues and spectral singularities

Principal functions