Self-adjoint extensions and spectral analysis in Calogero problem

TL;DR

This paper provides a rigorous quantum-mechanical analysis of the Calogero problem, exploring self-adjoint extensions, spectral properties, and symmetry breaking to improve understanding of singular potentials.

Contribution

It introduces a uniform approach to constructing self-adjoint Hamiltonians for the Calogero potential, addressing paradoxes and analyzing spectral and symmetry aspects.

Findings

Complete classification of self-adjoint Hamiltonians for the Calogero potential

Identification of spontaneous scale-symmetry breaking in certain extensions

Resolution of paradoxes in naive quantum treatments

Abstract

In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential $\alpha x^{-2}$. Although the problem is quite old and well-studied, we believe that our consideration, based on a uniform approach to constructing a correct quantum-mechanical description for systems with singular potentials and/or boundaries, proposed in our previous works, adds some new points to its solution. To demonstrate that a consideration of the Calogero problem requires mathematical accuracy, we discuss some "paradoxes" inherent in the "naive" quantum-mechanical treatment. We study all possible self-adjoint operators (self-adjoint Hamiltonians) associated with a formal differential expression for the Calogero Hamiltonian. In addition, we discuss a spontaneous scale-symmetry breaking associated with self-adjoint extensions. A complete spectral analysis of all self-adjoint Hamiltonians is presented.