SUSY Quantum Mechanics, (non)-Analyticity and $\ldots$ Phase Transitions

TL;DR

This paper investigates how discontinuities in the coupling constant of 1D Schrödinger equations lead to different types of phase transition-like behaviors in energies and eigenfunctions, using SUSY Quantum Mechanics as a framework.

Contribution

It classifies the types of discontinuities in energies and eigenfunctions in SUSY Quantum Mechanics, revealing three distinct phase transition-like behaviors.

Findings

Discontinuities can occur in energies and eigenfunctions due to coupling constant changes.

Three types of discontinuities are identified: first order, second order, and infinite order phase transition analogs.

Supersymmetric Quantum Mechanics effectively models these phenomena.

Abstract

It is shown by analyzing the $1D$ Schr\"odinger equation that discontinuities in the coupling constant can occur in both the energies and the eigenfunctions. Surprisingly, those discontinuities, which are present in the energies {\it versus} the coupling constant, are of three types only: (i) discontinuous energies (similar to 1st order phase transitions), (ii) discontinuous first derivative in the energy while the energy is continuous (similar to 2nd order phase transitions), (iii) the energy and all its derivatives are continuous but the functions are different below and above the point of discontinuity (similar to infinite order phase transitions). Supersymmetric (SUSY) Quantum Mechanics provides a convenient framework to study this phenomenon.