Exceptional polynomials and SUSY quantum mechanics

Chaitanya, K. V. S. Shiv; Ranjani, S. Sree; Panigrahi, Prasanta K.; Radhakrishnan, R.; Srinivasan, V.

doi:10.1007/s12043-014-0882-7

Cited by 10 publications

(22 citation statements)

References 17 publications

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“…Thus it leads to different kind of SUSY between the two partners. This is called as nontrivial supersymmetry [18]. Similar situations were encountered in the past, where isospectral deformation lead to non-shape invariant partners, but in these cases only few eigenfunctions could be calculated analytically for the deformed partner [37], [42].…”

Section: Nontrivial Supersymmetry and Conventional Supersymmetrymentioning

confidence: 91%

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Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials

Ranjani

2019

Pramana - J Phys

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A method to construct multi-indexed exceptional Laguerre polynomials using isospectral deformation technique and quantum Hamilton-Jacobi (QHJ) formalism is presented. We construct generalized superpotentials using singularity structure analysis which lead to rational potentials with multi-indexed polynomials as solutions. We explicitly construct such rational extensions of the radial oscillator and their solutions, which involve exceptional Laguerre orthogonal polynomials having two indices. The exact expressions for the L1, L2 and L3 type polynomials, along with their weight functions are presented. We also discuss the possibility of constructing more rational potentials with interesting solutions.keywords Exceptional orthogonal polynomials, multi-indexed exceptional polynomials, exactly solvable models, rational potentials, shape invariance, isospectral deformation, quantum Hamilton-Jacobi formalism, quantum momentum function. 1

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Section: Nontrivial Supersymmetry and Conventional Supersymmetrymentioning

confidence: 91%

“…Using the above equations, we can construct an hierarchy of ES potentials and their solutions from V − (x) and its solutions. The existence of the rational potentials has led to the idea of SUSY being nontrivial [18], which will be elaborated in the later sections.…”

Section: Supersymmetric Quantum Mechanicsmentioning

confidence: 99%

Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials

Ranjani

2019

Pramana - J Phys

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“…The hypergeometric polynomials,

({}, p_{n} ((), x))

, orthogonal with respect to the weight function

h ((), x)

on the support interval

normalΛ

, are known to often control the quantum‐mechanical wavefunctions of the bound states in numerous quantum systems [14, 39–43]. The Hermite, Laguerre and Jacobi polynomials are the three canonical families of real HOPs [44–47].…”

Section: Introductionmentioning

confidence: 99%

Algebraic and complexity‐like properties of Jacobi polynomials: Degree and parameter asymptotics

Sobrino

Dehesa

2021

Int J of Quantum Chemistry

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The Jacobi polynomials Pfalse^nαβ()x conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two‐parameter weight function 1−xα1+xβ,α,β>−1, on the interval [],−1+1. The spreading of its associated probability density (i.e., the Rakhmanov density) over the support interval has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic Lq‐norms (Shannon and Rényi entropies) and the monotonic complexity‐like measures of Cramér–Rao, Fisher–Shannon, and LMC (López‐Ruiz, Mancini, and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non‐handy way; specially when the degree or the parameters (),αβ are large. In this work, we determine in a simple, compact form the leading term of the entropic and complexity‐like properties of the Jacobi polynomials in the two extreme situations: (n→∞; fixed α,β) and (α→∞; fixed n,β). These two asymptotics are relevant per se and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of many exactly, conditional exactly, and quasi‐exactly solvable quantum‐ mechanical potentials which model numerous atomic and molecular systems.

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“…[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] Different methods like the Darboux-Crum method, 9,10 finite difference Backlünd algorithm, [11][12][13] prepotential method 14 and several others 15,16 were employed. Various groups explored the mathematical properties of the new polynomials concurrently.…”

Section: Introductionmentioning

confidence: 99%

“…Various groups explored the mathematical properties of the new polynomials concurrently. [19][20][21] Many studies showed that the traditional SIPs and their rational extensions are isospectral and share a supersymmetric partnership 3,15,16 and that the superpotentials of the traditional SIPs play a pivotal role. Now that the dust has settled over the construction of these new potentials, we think there is a need for a method, which is transparent, systematic and elegant.…”

Section: Introductionmentioning

confidence: 99%

Shape invariant rational extensions and potentials related to exceptional polynomials

Ranjani

Sandhya

Kapoor

2015

Int. J. Mod. Phys. A

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In this paper, we show that an attempt to construct shape invariant extensions of a known shape invariant potential leads to, apart from a shift by a constant, the well known technique of isospectral shift deformation. Using this, we construct infinite sets of generalized potentials with Xm exceptional polynomials as solutions. The method is simple and transparent and is elucidated using the radial oscillator and the trigonometric Pöschl-Teller potentials. For the case of radial oscillator, in addition to the known rational extensions, we construct two infinite sets of rational extensions, which seem to be less studied. Explicit expressions of the generalized infinite set of potentials and the corresponding solutions are presented. For the trigonometric Pöschl-Teller potential, our analysis points to the possibility of several rational extensions beyond those known in literature.

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Exceptional polynomials and SUSY quantum mechanics

Cited by 10 publications

References 17 publications

Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials

Quantum Hamilton–Jacobi route to exceptional Laguerre polynomials and the corresponding rational potentials

Algebraic and complexity‐like properties of Jacobi polynomials: Degree and parameter asymptotics

Shape invariant rational extensions and potentials related to exceptional polynomials

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