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

Alexander V Turbiner

Published 2024-09-04Version 1

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