On the self-adjointness of certain reduced Laplace-Beltrami operators

TL;DR

This paper investigates conditions under which reduced Laplace-Beltrami operators on complete Riemannian manifolds are self-adjoint, providing a concise criterion that applies to spin Calogero-Sutherland models with compact Lie group symmetries.

Contribution

It introduces a simple sufficient condition ensuring self-adjointness inheritance for reduced operators, simplifying the analysis of spin Calogero-Sutherland type Hamiltonians.

Findings

Provides a criterion for self-adjointness inheritance

Applies to spin Calogero-Sutherland models

Ensures self-adjointness of reduced Laplace-Beltrami operators

Abstract

The self-adjointness of the reduced Hamiltonian operators arising from the Laplace-Beltrami operator of a complete Riemannian manifold through quantum Hamiltonian reduction based on a compact isometry group is studied. A simple sufficient condition is provided that guarantees the inheritance of essential self-adjointness onto a certain class of restricted operators and allows us to conclude the self-adjointness of the reduced Laplace-Beltrami operators in a concise way. As a consequence, the self-adjointness of spin Calogero-Sutherland type reductions of `free' Hamiltonians under polar actions of compact Lie groups follows immediately.