Completeness theorem for the system of eigenfunctions of the complex Schr\"odinger operator $\mathscr{L}_c=-d^2/dx^2+cx^\alpha$

TL;DR

This paper proves the completeness of eigenfunctions for a class of complex Schrödinger operators on the semi-axis, expanding understanding of spectral properties under specific conditions on parameters.

Contribution

It establishes the completeness theorem for eigenfunctions of the complex Schrödinger operator with power-law potential for a range of parameters, extending previous spectral analysis.

Findings

Completeness holds for $orall \, ext{parameters}$ satisfying specified conditions.

The proof covers cases with complex potentials within certain argument bounds.

Results contribute to spectral theory of non-self-adjoint operators.

Abstract

The completeness of the system of eigenfunctions of the complex Schr\"odinger operator $\mathscr{L}_c=-d^2/dx^2+cx^\alpha$ on the semi-axis with Dirichlet boundary conditions is proved for $\alpha\in(0,2)$ and $|\arg c|<2\pi\alpha/(\alpha+2)+\Delta t(\alpha)$ with some $\Delta t(\alpha)>0$.