AN-Type Dunkl Operators and New Spin Calogero–Sutherland Models

A new family of A _N -type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians. Received: 17 February 2001 / Accepted: 8 March 2001     

The quasi-exactly solvable problems for two dimensional quantum systems

In this paper, we study the quasi-exactly solvable problems for two dimensional quantum systems. By using the Bethe ansatz method, we obtain the general form of the quasi-exactly solvable potential. Then, we present several examples to give the specific forms of quasi-exactly solvable potentials. In the examples, some physical models of quasi-exactly solvable problems are re-exhibited.     

Quasi-exact solvability beyond the <Emphasis Type="Italic">sl</Emphasis>(2) algebraization

We present evidence to suggest that the study of one-dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual sl(2) approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the sl(2) Liealgebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic Hamiltonian cannot be expressed as a polynomial in the generators of sl(2). We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie algebraic approach. The text was submitted by the authors in English.     

The Quasi-Exactly Solvable Problems in Relativistic Quantum Mechanics

We study the quasi-exactly solvable problems in relativistic quantum mechanics. We consider the problems for the two-dimensional Klein-Gordon and Dirac equations with equal vector and scalar potentials, and try to find the general form of the quasi-exactly solvable potential. After obtaining the general form of the potential, we present several examples to give the specific forms. In the examples, we show for special parameters the harmonic potential plus Coulomb potential, Killingbeck potential and a quartic potential plus Cornell potential are quasi-exactly solvable potentials.     

New quasi-exactly solvable Hamiltonians in two dimensions

Quasi-exactly solvable Schrödinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods. Such operators lie in the enveloping algebra of a finite-dimensional Lie algebra of first order differential operators—the hidden symmetry algebra. In this paper we develop some general techniques for constructing quasi-exactly solvable operators. Our methods are applied to provide a wide variety of new explicit two-dimensional examples (on both flat and curved spaces) of quasi-exactly solvable Hamiltonians, corresponding to both semisimple and more general classes of Lie algebras.Supported in Part by DGICYT Grant PS 89-0011.Supported in Part by an NSERC Grant.Supported in Part by NSF Grant DMS 92-04192.     

An approach to one-dimensional elliptic quasi-exactly solvable models

One-dimensional Jacobian elliptic quasi-exactly solvable second-order differential equations are obtained by introducing the generalized third master functions. It is shown that the solutions of these differential equations are generating functions for a new set of polynomials in terms of energy with factorization property. The roots of these polynomials are the same as the eigenvalues of the differential equations. Some one-dimensional elliptic quasi-exactly quantum solvable models are obtained from these differential equations.      

Quasi-exact solvability of Dirac-Pauli equation and generalized Dirac oscillators

In this paper we demonstrate that neutral Dirac particles in external electric fields, which are equivalent to generalized Dirac oscillators, are physical examples of quasi-exactly solvable systems. Electric field configurations permitting quasi-exact solvability of the system based on the sl(2) symmetry are discussed separately in the spherical, cylindrical, and Cartesian coordinates. Some exactly solvable field configurations are also exhibited.     

Application of asymptotic iteration method to the Khare-Mandal and its PT-symmetric QES partner potential

We study the application of the asymptotic iteration method to the Khare-Mandal potential and its PT-symmetric partner. The eigenvalues and eigenfunctions for both potentials are obtained analytically. We have shown that although the quasi-exactly solvable energy eigenvalues of the Khare-Mandal potential are found to be in complex conjugate pairs for certain values of potential parameters, its PT-symmetric partner exhibits real energy eigenvalues in all cases.      

Quantum Dynamical Algebra SU(1,1) in One-Dimensional Exactly Solvable Potentials

In this paper, we establish the underlying quantum dynamical algebra SU(1,1) for some one-dimensional exactly solvable potentials by using the shift operators method. The connection between SU(1,1) algebra and the radial Hamiltionian problems is also discussed. PACS numbers: 03.65.Ge     

A novel quasi-exactly solvable spin chain with nearest-neighbors interactions

In this paper we study a novel spin chain with nearest-neighbors interactions depending on the sites coordinates, which in some sense is intermediate between the Heisenberg chain and the spin chains of Haldane–Shastry type. We show that when the number of spins is sufficiently large both the density of sites and the strength of the interaction between consecutive spins follow the Gaussian law. We develop an extension of the standard freezing trick argument that enables us to exactly compute a certain number of eigenvalues and their corresponding eigenfunctions. The eigenvalues thus computed are all integers, and in fact our numerical studies evidence that these are the only integer eigenvalues of the chain under consideration. This fact suggests that this chain can be regarded as a finite-dimensional analog of the class of quasi-exactly solvable Schrödinger operators, which has been extensively studied in the last two decades. We have applied the method of moments to study some statistical properties of the chain's spectrum, showing in particular that the density of eigenvalues follows a Wigner-like law. Finally, we emphasize that, unlike the original freezing trick, the extension thereof developed in this paper can be applied to spin chains whose associated dynamical spin model is only quasi-exactly solvable.     

The multidimensional Darboux transformation

A generalization of the classical one-dimensional Darboux transformation to arbitrary n-dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n ≥ 2. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results.     

Quasi-exact solvability of Dirac equation with Lorentz scalar potential

We consider exact/quasi-exact solvability of Dirac equation with a Lorentz scalar potential based on factorizability of the equation. Exactly solvable and sl (2)-based quasi-exactly solvable potentials are discussed separately in Cartesian coordinates for a pure Lorentz potential depending only on one spatial dimension, and in spherical coordinates in the presence of a Dirac monopole.     

Prepotential approach to quasinormal modes

In this paper we demonstrate how the recently reported exactly and quasi-exactly solvable models admitting quasinormal modes can be constructed and classified very simply and directly by the newly proposed prepotential approach. These new models were previously obtained within the Lie-algebraic approach. Unlike the Lie-algebraic approach, the prepotential approach does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing, and gives the potential as well as the eigenfunctions and eigenvalues simultaneously. We also present three new models with quasinormal modes: a new exactly solvable Morse-like model, and two new quasi-exactly solvable models of the Scarf II and generalized Pöschl–Teller types.     

Generalization of quasi-exactly solvable and isospectral potentials

A unified approach in the light of supersymmetric quantum mechanics (SSQM) has been suggested for generating multidimensional quasi-exactly solvable (QES) potentials. This method provides a convenient means to construct isospectral potentials of derived potentials.      

Prepotential approach to exact and quasi-exact solvabilities

Exact and quasi-exact solvabilities of the one-dimensional Schrödinger equation are discussed from a unified viewpoint based on the prepotential together with Bethe ansatz equations. This is a constructive approach which gives the potential as well as the eigenfunctions and eigenvalues simultaneously. The novel feature of the present work is the realization that both exact and quasi-exact solvabilities can be solely classified by two integers, the degrees of two polynomials which determine the change of variable and the zeroth order prepotential. Most of the well-known exactly and quasi-exactly solvable models, and many new quasi-exactly solvable ones, can be generated by appropriately choosing the two polynomials. This approach can be easily extended to the constructions of exactly and quasi-exactly solvable Dirac, Pauli, and Fokker-Planck equations.     

An Alternative Method for Calculating Bound-State of Energy Eigenvalues of Klein-Gordon for Quasi-exactly Solvable Potentials

We obtain the bound-state energy of the Klein-Gordon equation for some examples of quasi-exactly solvable potentials within the framework of asymptotic iteration method （AIM）. The eigenvalues are calculated for type- 1 solutions. The whole quasi-exactly solvable potentials are generated from the defined relation between the vector and scalar potentials.     

Isospectral Hamiltonians from Moyal products

Recently Scholtz and Geyer proposed a very efficient method to compute metric operators for non-Hermitian Hamiltonians from Moyal products. We develop these ideas further and suggest to use a more symmetrical definition for the Moyal products, because they lead to simpler differential equations. In addition, we demonstrate how to use this approach to determine the Hermitian counterpart for a pseudo-Hermitian Hamiltonian. We illustrate our suggestions with the explicitly solvable example of the −x ⁴-potential and the ubiquitous harmonic oscillator in a complex cubic potential.     

Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

We completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi-exactly solvable Schrödinger operator on the line. Methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi-exactly solvable Hamiltonians and explicit normalizability conditions in general coordinate systems.Supported in Part by DGICYT Grant PS 89-0011Supported in Part by an NSERC GrantSupported in Part by NSF Grant DMS 92-04192     

Nonlinear supersymmetry on the plane in magnetic field and quasi-exactly solvable systems

The nonlinear n-supersymmetry with holomorphic supercharges is investigated for the 2D system describing the motion of a charged spin-1/2 particle in an external magnetic field. The universal algebraic structure underlying the holomorphic n-supersymmetry is found. It is shown that the essential difference of the 2D realization of the holomorphic n-supersymmetry from the 1D case recently analysed by us consists in appearance of the central charge entering non-trivially into the superalgebra. The relation of the 2D holomorphic n-supersymmetry to the 1D quasi-exactly solvable (QES) problems is demonstrated by means of the reduction of the systems with hyperbolic or trigonometric form of the magnetic field. The reduction of the n-supersymmetric system with the polynomial magnetic field results in the family of the one-dimensional QES systems with the sextic potential. Unlike the original 2D holomorphic supersymmetry, the reduced 1D supersymmetry associated with x⁶+⋯ family is characterized by the non-holomorphic supercharges of the special form found by Aoyama et al.