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.
Background
Laguerre-reference (LRef) and JRef are the two implicit potential families that are exactly solvable in terms of confluent hypergeometric or hypergeometric functions. These potentials are referred to as LRef and JRef potentials as they are quantized through classical Laguerre and classical Jacobi polynomials, respectively, using degree-dependent indices.
The JRef potential can be converted into a distinct rational function, referred to as the CGK potential, by the Darboux transformation with a nodeless eigenfunction as the transformation function. This implies that the exactly solvable potential like JRef is not shape-invariant, contradicting the renowned assertion of Gendenshtein that all exactly solvable potentials must be shape-invariant.
SUSY Rule Violation
This paper revealed some significant form-invariance features of the JRef canonical Sturm-Liouville equation (CSLE) in the particular density function case with the simple pole at the origin. It was proven that the CSLE preserves its form under the two second-order Darboux-Crum transformations (DCTs), with specially chosen basic quasi-rational solutions (q-RSs) pairs representing the seed functions to ensure that their analytical continuations do not possess zeros in the complex plane.
Additionally, both transformations often decrease or increase by two the exponent difference (ExpDiff) for the pole mentioned while keeping the other two parameters unchanged. The change can be more complicated if the ExpDiff for the original CSLE's pole is smaller than two at the origin. Researchers observed that bound energy levels are not preserved by DCTs according to conventional SUSY rules.
Specifically, they concluded that first-order differential expressions are generated by the mentioned second-order DCTs in the hypergeometric functions' space after the replacement of Crum Wronskians (CWs) by the Krein determinants (KDs). The predicted differential equations for the principal Frobenius solutions (PFSs) near the origin were explicitly confirmed using the conventional contiguous relations for hypergeometric series.
This mentioned anomalous case presented a good example of the breakup of the conventional SUSY quantum mechanics rules for DCTs between limit point (LP) and limit circle (LC) regions.
The Study and Findings
In this study, the researchers split the DCT into the two sequential Darboux deformations of the Liouville potentials associated with the CSLEs to understand the anomaly. Specifically, the anomaly source was explained by decomposing the second-order DCT into two sequential Liouville-Darboux transformations (LDTs).
The two different Liouville transformations on the infinite interval (1, inf) and on the finite interval (0, 1) then resulted in the supplementary double-step shape-invariant potential pair defined on the real axis and on the positive semi-axis, respectively. These potentials were solvable by the Heun equation's polynomial solutions.
The researchers observed that the initial CSLE was turned into the Heun equation written in canonical form by the first Darboux transformation, while the second transformation yielded the hypergeometric equation's canonical form. Additionally, the first of these transformations/first LDT placed the ExpDiff into the LC range, and then the second transformation/second LDT kept the pole/the given spectral problem within the LC region, which violated the conventional SUSY quantum mechanics prescriptions.
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 RCSLE is referred to as Routh-reference (RRef), LRef, or JRef if it possesses q-RSs consisting of generalized Routh, Jacobi, or Laguerre polynomials. Most importantly, the eigenfunctions of the LRef and JRef CSLEs are composed of infinite sequences of classical Laguerre and classical Jacobi polynomials, with the polynomial indices typically reliant on the polynomial degrees in all three cases, while the RRef CSLE is quantized based on Romanovski/pseudo-Jacobi polynomials' finite orthogonal sequences.
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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