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

Alexander V Turbiner

TL;DR

The paper demonstrates that in one-dimensional quantum mechanics, energies and eigenfunctions can exhibit non-analytic, piecewise dependence on coupling constants, a phenomenon illuminated through SUSY quantum mechanics. By analyzing several SUSY-based potentials, including $w=\omega x$, $w=a x^3$, and generalized QES sextic cases, it identifies three distinct discontinuity types in $E(a)$ (first-, second-, and infinite-order), governed by the signs and combinations of parameters such as $a$ and $b$. The work combines exact SUSY relations, Bohr–Sommerfeld quantization, WKB corrections $\gamma(N)$, and instanton-like expansions to map out the analytic structure of the spectra and shows that confinement to a finite box removes these discontinuities. The findings illuminate a phase-transition–like landscape in 1D QM and hint at deeper connections between SUSY, quasi-exact solvability, and non-analytic spectral behavior with potential implications for understanding SUSY breaking and non-perturbative effects.

Abstract

It is shown by analyzing the $1D$ Schrödinger 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.

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

TL;DR

The paper demonstrates that in one-dimensional quantum mechanics, energies and eigenfunctions can exhibit non-analytic, piecewise dependence on coupling constants, a phenomenon illuminated through SUSY quantum mechanics. By analyzing several SUSY-based potentials, including , , and generalized QES sextic cases, it identifies three distinct discontinuity types in (first-, second-, and infinite-order), governed by the signs and combinations of parameters such as and . The work combines exact SUSY relations, Bohr–Sommerfeld quantization, WKB corrections , and instanton-like expansions to map out the analytic structure of the spectra and shows that confinement to a finite box removes these discontinuities. The findings illuminate a phase-transition–like landscape in 1D QM and hint at deeper connections between SUSY, quasi-exact solvability, and non-analytic spectral behavior with potential implications for understanding SUSY breaking and non-perturbative effects.

Abstract

It is shown by analyzing the Schrödinger 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.
Paper Structure (10 sections, 55 equations, 13 figures)

This paper contains 10 sections, 55 equations, 13 figures.

Table of Contents

  1. Introduction
  2. SUSY Quantum Mechanics, see e.g. SUSY:1995
  3. $w\ =\ \omega x$ - Harmonic oscillator
  4. $w\ =\ a x^3$ - primitive quasi-exactly-solvable (QES) sextic oscillator, see Turbiner:1988
  5. $w\ =\ a x^2$ - SUSY- broken quartic oscillator, $\Psi_0 \notin L_2(R)$, $E_B^{(0)}\neq 0$
  6. $w\ =\ a x^3 + b x$ - generalized primitive QES sextic polynomial oscillator Turbiner:1988
  7. $w=a x^2 + b x$ - primitive, SUSY-broken, quartic anharmonic oscillator
  8. Exact Bohr-Sommerfeld (B-S) quantization condition and SUSY quantum mechanics
  9. WKB correction $\gamma$ for SUSY potentials $x^{6} \mp 3 x^2$
  10. Conclusions

Figures (13)

  • Figure 1: Ground state energy of the harmonic oscillator vs. frequency $\omega$.
  • Figure 2: Ground state energy of the harmonic oscillator in the box of size $2L$vs. frequency $\omega$: an approximate expression.
  • Figure 3: The harmonic oscillator: Ground state energy $E_0$vs. frequency $\omega$
  • Figure 4: Bosonic potential $V_B(x; a)$ (\ref{['VB6']}) at $a>0$, where the ground state energy $E_0=0$ for any $a>0$
  • Figure 5: Bosonic potential $V_B(x; a)$ (\ref{['VB6']}) at $a < 0$, where the ground state energy $E_0 \neq 0$ for any $a<0$
  • ...and 8 more figures