[Paper Review] Comment on "Self-Isospectral Periodic Potentials and Supersymmetric Quantum Mechanics"

U. Sukhatme, Avinash Khare|ArXiv.org|Feb 23, 1999

Quantum Mechanics and Non-Hermitian Physics1 references3 citations

TL;DR

This paper demonstrates that applying supersymmetric quantum mechanics (SUSY QM) to solvable elliptic function potentials, specifically $ V(x) = mj(j+1)\text{sn}^2(x,m) $, generates new exactly solvable one-dimensional periodic potentials. The method leverages SUSY transformations to construct isospectral partners, yielding novel periodic systems with analytically tractable energy spectra.

ABSTRACT

We show that the formalism of supersymmetric quantum mechanics applied to the solvable elliptic function potentials $V(x) = mj(j+1){sn}^2(x,m)$ produces new exactly solvable one-dimensional periodic potentials.

Motivation & Objective

• To explore the application of supersymmetric quantum mechanics (SUSY QM) to self-isospectral periodic potentials.
• To investigate whether SUSY transformations on known solvable periodic potentials yield new exactly solvable systems.
• To extend the class of exactly solvable periodic potentials using SUSY QM formalism.
• To analyze the spectral properties of the resulting potentials and confirm their isospectrality.
• To contribute to the theoretical framework of supersymmetric quantum mechanics in periodic systems.

Proposed method

• Application of supersymmetric quantum mechanics (SUSY QM) to the known solvable potential $ V(x) = mj(j+1)\text{sn}^2(x,m) $, where $ \text{sn}(x,m) $ is the Jacobi elliptic function.
• Use of the standard SUSY QM formalism involving partner Hamiltonians $ H_+ $ and $ H_- $, constructed via a superpotential derived from the ground state wavefunction.
• Derivation of the partner potential $ V_+(x) $ using the standard SUSY transformation formula $ V_+(x) = V_-(x) + 2 \frac{d^2}{dx^2} \ln \psi_0(x) $.
• Verification of isospectrality between the original and transformed potentials through analytical computation of energy levels.
• Utilization of properties of elliptic functions and their derivatives to ensure periodicity and solvability of the new potential.
• Analysis of the resulting potential's structure to confirm it is a new, non-trivial, exactly solvable periodic potential.

Experimental results

Research questions

• RQ1 Can supersymmetric quantum mechanics generate new exactly solvable periodic potentials from known solvable ones?
• RQ2 What is the structure of the partner potential obtained via SUSY transformation of the $ \text{sn}^2 $-type potential?
• RQ3 Are the new potentials isospectral to the original potential, and do they retain periodicity?
• RQ4 What are the analytical properties of the wavefunctions and energy spectra of the new potentials?
• RQ5 How does the SUSY QM formalism extend the class of solvable periodic systems?

Key findings

• The SUSY QM transformation applied to $ V(x) = mj(j+1)\text{sn}^2(x,m) $ produces a new, non-trivial, exactly solvable one-dimensional periodic potential.
• The resulting partner potential is isospectral to the original potential, preserving all energy levels except possibly the ground state.
• The new potential maintains periodicity due to the underlying elliptic function structure and the properties of the superpotential.
• The method systematically extends the class of known exactly solvable periodic potentials in quantum mechanics.
• The energy spectrum of the new potential is analytically tractable, confirming its solvability.
• The construction demonstrates a general mechanism for generating new solvable periodic systems from known ones via SUSY QM.