Spectral properties of non-selfadjoint extensions of the Calogero Hamiltonian

TL;DR

This paper characterizes all non-selfadjoint extensions of the Calogero Hamiltonian that have a non-empty resolvent and generate an analytic semigroup, focusing on spectral properties relevant to quantum mechanics.

Contribution

It provides a complete description of non-selfadjoint extensions of the Calogero Hamiltonian with specific spectral and semigroup generation properties.

Findings

Identifies all extensions with non-empty resolvent

Characterizes those generating an analytic semigroup

Analyzes spectral properties of these extensions

Abstract

We describe all extensions of the Calogero Hamiltonian \[L=-\frac{d^2}{dr^2}+\frac{b}{r^2} \quad \text{in}\ L^2(\mathbb{R}_+), \quad b <-\frac{1}{4}\] having non empty resolvent and generating an analytic semigroup in $L^2(\mathbb{R}_+)$.