Singular Eigenfunctions of Calogero-Sutherland Type Systems and How to Transform Them into Regular Ones

TL;DR

This paper explores a class of Calogero-Sutherland type quantum systems with variable masses and couplings, demonstrating how to transform their singular eigenfunctions into regular, physically acceptable solutions using integral operators.

Contribution

It introduces a method to convert singular eigenfunctions of generalized Calogero-Sutherland systems into regular ones via integral transformations.

Findings

Existence of explicit eigenfunctions for variable-mass Calogero-Sutherland systems.

Transformation technique for converting singular eigenfunctions into regular, square-integrable ones.

Connection between non-Hermitian and Hermitian Hamiltonians in these models.

Abstract

There exists a large class of quantum many-body systems of Calogero-Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exact eigenfunctions and corresponding eigenvalues, explicitly. Of course there is a catch to this result: if one insists on these eigenfunctions to be square integrable then the corresponding Hamiltonian is necessarily non-hermitean (and thus provides an example of an exactly solvable PT-symmetric quantum-many body system), and if one insists on the Hamiltonian to be hermitean then the eigenfunctions are singular and thus not acceptable as quantum mechanical eigenfunctions. The standard Calogero-Sutherland Hamiltonian is special due to the existence of an integral operator which allows to transform these singular eigenfunctions into regular ones.