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Double-Step Shape Invariance in Quantum Mechanics

An article recently published in the journal Axioms explored the double-step shape invariance of radial Jacobi-Reference (JRef) potential and the violation of conventional supersymmetric (SUSY) quantum mechanics rules.

Double-Step Shape Invariance in Quantum Mechanics
Study: Double-Step Shape Invariance in Quantum Mechanics. Image Credit: metamorworks/


Laguerre-reference (LRef) and Jacobi-reference (JRef) are two families of solvable potentials expressed in terms of confluent hypergeometric or hypergeometric functions. These are known as LRef and JRef potentials because their quantization is achieved through classical Laguerre and Jacobi polynomials, respectively, with degree-dependent indices.

The JRef potential can be transformed into a distinct rational function known as the CGK potential through the Darboux transformation, utilizing a nodeless eigenfunction as the transformation function. This conversion suggests that the JRef potential, despite being exactly solvable, is not shape-invariant. This finding challenges Gendenshtein's well-known assertion that all exactly solvable potentials must maintain shape invariance.

SUSY Rule Violation

This study highlighted notable form-invariance traits of the JRef canonical Sturm-Liouville equation (CSLE) when utilizing a specific density function with a simple pole at the origin. It was demonstrated that the CSLE maintains its form under two second-order Darboux-Crum transformations (DCTs), using specially selected pairs of basic quasi-rational solutions (q-RSs) as seed functions. These solutions ensure their analytical continuations remain zero-free in the complex plane.

Furthermore, these transformations typically alter the exponent difference (ExpDiff) associated with the pole by two units while maintaining the stability of other parameters. However, if the original CSLE's pole at the origin has an ExpDiff of less than two, the transformation effects become more complex. It was observed that the bound energy levels are not conserved by DCTs as per traditional SUSY rules.

Remarkably, the study revealed that these second-order DCTs generate first-order differential expressions within the space of hypergeometric functions after substituting Crum Wronskians (CWs) with Krein determinants (KDs). The resulting differential equations for the principal Frobenius solutions (PFSs) near the origin were conclusively verified using standard contiguous relations for hypergeometric series.

This particular case serves as a compelling example of how conventional SUSY quantum mechanics rules are disrupted when DCTs are applied between limit point (LP) and limit circle (LC) conditions.

The Study and Findings

In this research, the team dissected the DCT into two sequential Darboux deformations of the Liouville potentials related to the CSLEs to pinpoint the source of the observed anomaly. Specifically, they explained the anomaly by breaking down the second-order DCT into two consecutive Liouville-Darboux transformations (LDTs).

The researchers applied two different Liouville transformations—one on the infinite interval (1, ∞) and another on the finite interval (0, 1). This approach generated a pair of supplementary double-step shape-invariant potentials defined respectively on the real axis and the positive semi-axis. These potentials were solvable using the polynomial solutions of the Heun equation.

The study revealed that the initial CSLE was transformed into the canonical form of the Heun equation by the first Darboux transformation. The second transformation subsequently converted it into the canonical form of the hypergeometric equation. Furthermore, the first transformation shifted the ExpDiff into the LC range, and the second transformation maintained the pole within the LC region. This sequence of transformations contradicted traditional SUSY quantum mechanics rules, illustrating a significant deviation from expected outcomes.

Significance of the Work

The significance of this study extends beyond the specific findings outlined here. It presented a specific illustration of the recently developed SUSY theory of the Gauss-reference (GRef) potentials representing the Liouville potentials for the confluent rational CSLE (RCSLE) with a single pole in the finite plane that is commonly placed at the origin or two Fuchsian RCSLEs with three second-order poles, including infinity.

The CSLE is categorized as Routh-reference (RRef), LRef, or JRef, depending on the types of q-RSs it incorporates. These solutions consist of generalized Routh, Jacobi, or Laguerre polynomials, respectively. Crucially, the eigenfunctions for the LRef and JRef CSLEs are comprised of infinite sequences of classical Laguerre and Jacobi polynomials, with the indices of these polynomials typically dependent on their degrees. In contrast, the RRef CSLE utilizes finite orthogonal sequences of Romanovski or pseudo-Jacobi polynomials for quantization.

In this work, the form-invariant RCSLE concept was extended to the JRef CSLE with the density function. The Liouville transformation can be performed independently on the three quantization intervals (−∞, 0), (1, ∞), and (0, 1), which leads to three Liouville potentials, including the radial potential studied in this work and the two branches of the linear tangent polynomial (LTP) potential on the line, which were found to be shape-invariant due to the action of second-order DCTs with basic solution pairs as the seed functions.

To summarize, this study effectively explained the breakdown of SUSY rules for the radial potential with a centrifugal barrier in the LC range.

Journal Reference

Natanson, G. (2024). Double-Step Shape Invariance of Radial Jacobi-Reference Potential and Breakdown of Conventional Rules of Supersymmetric Quantum Mechanics. Axioms, 13(4), 273.,

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Samudrapom Dam

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Samudrapom Dam

Samudrapom Dam is a freelance scientific and business writer based in Kolkata, India. He has been writing articles related to business and scientific topics for more than one and a half years. He has extensive experience in writing about advanced technologies, information technology, machinery, metals and metal products, clean technologies, finance and banking, automotive, household products, and the aerospace industry. He is passionate about the latest developments in advanced technologies, the ways these developments can be implemented in a real-world situation, and how these developments can positively impact common people.


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