Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

González‐López, A.; Kamran, Niky; Olver, Peter J.

doi:10.1007/bf02099042

Cited by 132 publications

(280 citation statements)

References 16 publications

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“…(5.34) of González-López, Kamran and Olver [18]; (ii) biconfluent Heun equations for the sextic potential V 1 z 6 + V 2 z 4 + V 3 z 2 + V 4 + V 5 /z 2 given by Turbiner [38] and Ushveridze [40], and for the potentials II, III and VIII given in Turbiner's list [38]; (iii) double-confluent Heun equations for the inverse fourth-power potential V (r) = V 1 r −4 + V 2 r −3 + V 3 r −2 + V 4 r −1 [37,38], and for the asymmetric doubleMorse potential given by Zaslavskii and Ulyanov [44]; (iv) confluent Heun equations for the trigonometric and hyperbolic potentials given by Ushveridze [39]; (v) general Heun equations in the Darboux elliptic form [9] for the first and second Ushveridze's elliptic potentials [39].…”

Section: Appendix a Heun Equations And Quasiexact Solvabilitymentioning

confidence: 99%

Solutions for confluent and double-confluent Heun equations

El-Jaick

Figueiredo

2008

Journal of Mathematical Physics

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Abstract. This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. Finally, we present problems which are ruled by each of these four equations and establish relations among Heun equations and quasi-exactly solvable problems.

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Section: Appendix a Heun Equations And Quasiexact Solvabilitymentioning

confidence: 99%

Solutions for confluent and double-confluent Heun equations

El-Jaick

Figueiredo

2008

Journal of Mathematical Physics

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“…These are called quasi-exactly solvable (QES) problems for which it is possible to determine analytically a part of the spectrum but not the whole spectrum [7,8,9,10,11,12]. The discovery of this class of spectral problems has greatly enlarged the number of physical systems which we can study analytically.…”

mentioning

confidence: 99%

“…Particularly, x = z dy/ P 4 (y). Analysis on the inequivalent forms of real quartic polynomials P 4 thus give a classification of all sl(2)-based QES Hamiltonians [8,9]. If one imposes the requirement of non-periodic potentials, then there are only five inequivalent classes, which are called case 1 to 5 in [9].…”

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confidence: 99%

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Quasi-exactly solvable quasinormal modes

Cho

2007

AIP Conference Proceedings

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We consider quasinormal modes with complex energies from the point of view of the theory of quasi-exactly solvable (QES) models. We demonstrate that it is possible to find new potentials which admit exactly solvable or QES quasinormal modes by suitable complexification of parameters defining the QES potentials. Particularly, we obtain one QES and four exactly solvable potentials out of the five one-dimensional QES systems based on the sl(2) algebra.

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The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials

Finkel

Kamran

1998

Advances in Applied Mathematics

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We prove that the scalar and 2 = 2 matrix differential operators which preserve the simplest scalar and vector-valued polynomial modules in two variables have a fundamental Lie algebraic structure. Our approach is based on a general graphical method which does not require the modules to be irreducible under the action of Ž . the corresponding Lie super algebra. This method can be generalized to modules of polynomials in an arbitrary number of variables. We given generic examples of partially solvable differential operators which are not Lie algebraic. We show that certain vector-valued modules give rise to new realizations of finite-dimensional Lie superalgebras by first-order differential operators. ᮊ

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Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

Cited by 132 publications

References 16 publications

Solutions for confluent and double-confluent Heun equations

Solutions for confluent and double-confluent Heun equations

Quasi-exactly solvable quasinormal modes

The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials

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