Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics

Quesne, Christiane

doi:10.1088/1742-6596/380/1/012016

Cited by 14 publications

(14 citation statements)

References 94 publications

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“…In the literature, exceptional Laguerre polynomials are often denoted as

X_{m}

‐Laguerre polynomials where m is the number of exceptional degrees . In our set‐up, this number of exceptional degrees is given by

m = false| {double-struckN}_{0} ∖ {double-struckN}_{λ, μ} false| = false| λ false| + false| μ false|,

where

N_{λ, μ}

is the degree sequence .…”

Section: Construction Of the Exceptional Laguerre Polynomialmentioning

confidence: 99%

Exceptional Laguerre Polynomials

Bonneux

Kuijlaars

2018

Stud Appl Math

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The aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials, we associate them with two partitions. We find that the use of partitions is an elegant way to express these polynomials and we restate some of their known properties in terms of partitions. We discuss the asymptotic behavior of the regular zeros and the exceptional zeros of exceptional Laguerre polynomials as the degree tends to infinity.

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“…In the literature, exceptional Laguerre polynomials are often denoted as

X_{m}

‐Laguerre polynomials where m is the number of exceptional degrees . In our set‐up, this number of exceptional degrees is given by

m = false| {double-struckN}_{0} ∖ {double-struckN}_{λ, μ} false| = false| λ false| + false| μ false|,

where

N_{λ, μ}

is the degree sequence .…”

Section: Construction Of the Exceptional Laguerre Polynomialmentioning

confidence: 99%

Exceptional Laguerre Polynomials

Bonneux

Kuijlaars

2018

Stud Appl Math

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“…It is indeed close to potentials widely used in molecular physics to describe out-of-plane bending vibrations and in solid state physics to provide models for one-dimensional crystals [3]. It is also related to the Scarf I potential [4] via simple changes of variable and of parameters [5].…”

Section: Introductionmentioning

confidence: 85%

Quasi-exactly solvable extended trigonometric Pöschl-Teller potentials with position-dependent mass

Quesne

2019

Eur. Phys. J. Plus

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Infinite families of quasi-exactly solvable position-dependent mass Schrödinger equations with known ground and first excited states are constructed in a deformed supersymmetric background. The starting points consist in one-and two-parameter trigonometric Pöschl-Teller potentials endowed with a deformed shape invariance property and, therefore, exactly solvable. Some extensions of them are considered with the same position-dependent mass and dealt with by a generating function method. The latter enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential from some generating function W + (x) [and its accompanying function W − (x)]. The generalized trigonometric Pöschl-Teller potentials so obtained are thought to have interesting applications in molecular and solid state physics.

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“…70 The problem (19)-(21) is known as the rationally extended Scarf model; 68,69 if the following additional constraints on the parameters a, b, and m are imposed:…”

Section: The Scarf I Systemmentioning

confidence: 99%

“…In this section, we will take up results found in Ref. 69 in order to convert them to the time-dependent scenario.…”

Section: The Radial Oscillator Systemmentioning

confidence: 99%