Special Issue: “Symmetries in Quantum Mechanics and Statistical Physics”

Junker, Georg

doi:10.3390/sym13112027

Symmetry

2021

DOI: 10.3390/sym13112027

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Special Issue: “Symmetries in Quantum Mechanics and Statistical Physics”

Georg Junker

Abstract: Symmetry is a fundamental concept in science and has played a significant role since the early days of quantum physics [...]

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Potentials from the Polynomial Solutions of the Confluent Heun Equation

Lévai

2023

Symmetry

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Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function Hc(p,β,γ,δ,σ;z). One of these recovers the generalized Laguerre polynomials LN(α), and another one the rationally extended X1 type Laguerre polynomials L^N(α). The two remaining solutions represent previously unknown polynomials that do not form an orthogonal set and exhibit features characteristic of semi-classical orthogonal polynomials. A standard method of generating exactly solvable potentials in the one-dimensional Schrödinger equation is applied to the CHE, and all known potentials with solutions expressed in terms of the generalized Laguerre polynomials within, or outside the Natanzon confluent potential class, are recovered. It is also found that the potentials generated from the two new polynomial systems necessarily depend on the N quantum number. General considerations on the application of the Heun type differential differential equations within the present framework are also discussed.

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Potentials from the Polynomial Solutions of the Confluent Heun Equation

Lévai

2023

Symmetry

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scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

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Special Issue: “Symmetries in Quantum Mechanics and Statistical Physics”

Abstract: Symmetry is a fundamental concept in science and has played a significant role since the early days of quantum physics [...]

Cited by 1 publication

References 8 publications

Potentials from the Polynomial Solutions of the Confluent Heun Equation

Potentials from the Polynomial Solutions of the Confluent Heun Equation

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