On the symmetry of commuting differential operators with singularities along hyperplanes

TL;DR

This paper investigates the symmetry properties of commuting differential operators with singular potentials along hyperplanes, revealing how inverse square singularities and coupling constants induce Weyl group symmetries.

Contribution

It demonstrates that inverse square singularities in potential functions lead to Weyl group symmetries in the commutants of Schrödinger operators, highlighting a natural connection.

Findings

Weyl group symmetry arises from potential singularities

Inverse square singularities influence operator commutants

Generic coupling constants support symmetry results

Abstract

We study the commutants of a Schr\"{o}dinger operator whose potential function possesses inverse square singularities along some hyperplanes passing through the origin. It is shown that the Weyl group symmetry of the potential function and the commutants naturally results from such singularities and the generic nature of the coupling constants.