Homogeneous Schr\"odinger operators on half-line

TL;DR

This paper investigates a family of Schrödinger operators with inverse-square potentials on the half-line, analyzing their spectral and scattering properties as the parameter varies.

Contribution

It introduces a holomorphic extension of the operators with respect to the parameter m and studies their spectral and scattering behavior.

Findings

Operators have a unique holomorphic extension for Re(m) > -1

Spectral properties depend smoothly on the parameter m

Scattering properties are characterized across the parameter range

Abstract

The differential expression $L_m=-\partial_x^2 +(m^2-1/4)x^{-2}$ defines a self-adjoint operator H_m on L^2(0;\infty) in a natural way when $m^2 \geq 1$. We study the dependence of H_m on the parameter m, show that it has a unique holomorphic extension to the half-plane Re(m) > -1, and analyze spectral and scattering properties of this family of operators.