On continuous movement of the discrete spectrum of Schr\"odinger operators

TL;DR

This paper investigates how the discrete spectrum of a Schr"odinger operator evolves continuously as a complex parameter varies, providing insights into spectral changes in non-selfadjoint operators derived from selfadjoint cases.

Contribution

It introduces a detailed analysis of the continuous movement of the discrete spectrum for a class of Schr"odinger operators with complex potentials.

Findings

Discrete spectrum members evolve continuously with parameter changes.

Provides conditions under which spectral members move along continuous paths.

Links spectral evolution to properties of the potential and operator.

Abstract

Continuous movement of discrete spectrum of the Schr\"{o}dinger operator $H(z)=-\frac{d^2} {dx^2}+V_0+z V_1$, with $\int_0^\infty {x |V_j(x)| dx} < \infty$, on the half-line is studied as $z$ moves along a continuous path in the complex plane. The analysis provides information regarding the members of the discrete spectrum of the non-selfadjoint operator that are evolved from the discrete spectrum of the corresponding selfadjoint operator.