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

Abstract

This study focuses on the spectral features of the non-hermitian singular operator with an out of the ordinary type weight function. Take into consideration the one-dimensional time dependent Schrödinger type differential equation -y^( ^'' )+q(x)y=μ^2 ρ(x)y,x∈[0,∞), holding the initial condition y(0)=0, and the density function defined with completely negative value as ρ(x)=-1. There are enormous number of the papers considering the positive values of ρ(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. Differently from the classical literature, we used the hyperbolic type representations of the fundamendal solutions of the equation to obtain the spectrum of the operator. Additionally, the requirements for finiteness of eigenvalues and spectral singularities were addressed. Hence, Naimark’s and Pavlov’s conditions were adopted to the negative density function case.