A Study on the Non-selfadjoint Schrödinger Operator with Negative Density Function

Authors : Nimet COSKUN

Pages : 220-229

Doi:10.53433/yyufbed.1139044

View : 9 | Download : 3

Publication Date : 2023-04-30

Article Type : Research Paper

Abstract :This study focuses on the spectral features of the non-selfadjoint singular operator with an out-of-the-ordinary type weight function. Take into consideration the one-dimensional time-dependent Schrödinger type differential equation -y^\`\`+qinsert ignore into journalissuearticles values(x);y=μ^2 ρinsert ignore into journalissuearticles values(x);y,x∈[0,∞);, holding the initial condition yinsert ignore into journalissuearticles values(0);=0, and the density function defined with a completely negative value as ρinsert ignore into journalissuearticles values(x);=-1. There is an enormous number of the papers considering the positive values of ρinsert ignore into journalissuearticles values(x); for both continuous and discontinuous cases. The structure of the density function affects the analytical properties and representations of the solutions of the equation. Unlike the classical literature, we use the hyperbolic type representations of the equation’s fundamental solutions to obtain the operator’s spectrum. Additionally, the requirements for finiteness of eigenvalues and spectral singularities are addressed. Hence, Naimark’s and Pavlov’s conditions are adopted for the negative density function case.
Keywords : Negatif yoğunluk fonksiyonu, Spektral analiz, Spektral tekillikler