Spectral singularities and quasi-exactly solvable quantal problem

Turbiner, A. V.; Ushveridze, A.G.

doi:10.1016/0375-9601(87)90456-7

Cited by 184 publications

(269 citation statements)

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“…Following Refs. [20] and [21], we shall call such models quasi-exactly solvable (QES). A somewhat different approach was followed by Hou and Shifman [22], who obtained a deformation of the B N Calogero model by studying the motions of the zeros of suitable solutions of the time-dependent Schrödinger equation of a well-known QES one-particle polynomial potential.…”

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confidence: 99%

Exact solutions of a new elliptic Calogero–Sutherland model

2001

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A quantum Hamiltonian describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is introduced. For a discrete set of values of the strength of the external potential, it is shown that a finite number of eigenfunctions and eigenvalues of the model can be exactly computed in an algebraic way.  2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Fd; 71.10.Pm; 11.10.Lm It is well known that the class of exactly solvable problems does not include most physical problems. The development of computer science in the last decades has made possible the use of numerical methods to approximate exact solutions in a wide variety of situations. Yet, the study of exactly solvable models still deserves attention, not only because the knowledge of exact solutions can be used to test approximate methods, but also in its own right, due to the simplicity and mathematical beauty of the models, and the wide range of connections with other fields of physical and mathematical research. This is illustrated by the renewed interest in the Calogero-Sutherland (CS) models of interacting particles in one dimension, which have been recently applied to many different fields such us quantum spin chains with long range interaction The first example of a non-trivial integrable quantum many-body problem was found by Calogero [8], and consists of a system of identical nonrelativistic particles interacting pairwise through an inverse-square potential v(r) = r −2 , so that the Hamiltonian is

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confidence: 99%

Exact solutions of a new elliptic Calogero–Sutherland model

2001

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“…The function g(x) plays the role of a weight factor, [f (x)] λ N is introduced due to eventual singularities in (2), and the quantities c (N ) m are expansion coefficients on the basis h(x). We introduce the notation…”

Section: A Qes General Approachmentioning

confidence: 99%

“…This however does not lead to new results: In fact, we just recover the case N = 0 and N = 1, respectively. This is seen as follows, taking the case N = 2 as an example: Using the constraints (56) and (57), we can solve the equation for c (2) 2 as a function of c (2) 0 . We get c …”

Section: The General Casementioning

confidence: 99%

A General Approach of Quasi-Exactly Solvable Schrödinger Equations

2002

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We construct a general algorithm generating the analytic eigenfunctions as well as eigenvalues of one-dimensional stationary Schrödinger Hamiltonians. Both exact and quasi-exact Hamiltonians enter our formalism but we focus on quasi-exact interactions for which no such general approach has been considered before. In particular we concentrate on a generalized sextic oscillator but also on the Lamé and the screened Coulomb potentials.

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“…Quasi-exactly solvable (QES) Hamiltonians, interconnecting a diverse array of physical problems, have been the subject of extensive study in recent times [1,2,3,4,5,6]. These systems, containing a finite number of exactly obtainable eigenstates, have been linked with classical electrostatic problems, as also to the finite dimensional irreducible representations of certain algebras.…”

Section: Introductionmentioning

confidence: 99%

“…These systems, containing a finite number of exactly obtainable eigenstates, have been linked with classical electrostatic problems, as also to the finite dimensional irreducible representations of certain algebras. Some of these studies employ group theoretical methods, others are based on the symmetry of the relevant differential equations [2,3,4]. The key to the existence of the finite number of identifiable states is the quasi-exact solvability condition, relating certain potential parameters of these dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Periodic Quasi-Exactly Solvable Models

2005

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Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lamé potential, are shown to emerge naturally in the quantum Hamilton-Jacobi approach. It is found that, the intrinsic nonlinearity of the Riccati type quantum Hamilton-Jacobi equation is primarily responsible for the surprisingly large number of allowed solvability conditions in the associated Lamé case. We also study the singularity structure of the quantum momentum function, which yields the band edge eigenvalues and eigenfunctions. * akksprs@uohyd.ernet.in † akksp@uohyd.ernet.in ‡ prasanta@prl.ernet.in

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Spectral singularities and quasi-exactly solvable quantal problem

Cited by 184 publications

References 6 publications

Exact solutions of a new elliptic Calogero–Sutherland model

Exact solutions of a new elliptic Calogero–Sutherland model

A General Approach of Quasi-Exactly Solvable Schrödinger Equations

Periodic Quasi-Exactly Solvable Models

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