Algebraic methods for the Natanzon potentials

It is shown that the Schrödinger equation can be solved by means of spectrum-generating algebra techniques for the most general class of Natanzon potentials based on the SO(2, 1) algebra. This paper describes in detail thelinear spectrum generating algebra method which is then applied to solve the Natanzon confluent potentials, and it is extended to one example with spin-orbit coupling. Further, the method is used to explain in detail how to find the energy spectrum for the Dirac equation with a Coulomb potential. Afterwards thequadratic spectrum generation algebra method is presented, and it is used to solve the most general hypergeometric Natanzon potential: The bound state problem and the corresponding wave functions are given. A simple example further illustrates the use of the quadratic method.Partially supported by grant FONDECYT 90-1240.     

两类新的条件精确可解势及其非线性谱生成代数

从平移的谐振子势出发,利用超对称量子力学构造出两类新的条件精确可解势,其中一类同时出现超对称性和空间对称性的破缺．此外,还构造出了这两类新可解势的非线性谱生成代数. 关键词：     

Liouville Transformation and Exactly Solvable Schrodinger Equations

The present paper discusses the connectionbetween exactly solvable Schrodinger equations and theLiouville transformation. This transformation yields alarge class of exactly solvable potentials, including the exactly solvable potentials introduced byNatanzon. In addition, this class is shown to containtwo new families of exactly solvablepotentials.     

A Group-Theoretical Method for Natanzon Potentials in Position-Dependent Mass Background by Means of Conformal Mappings

A new manner for deriving the exact potentials is presented. By making use of conformal mappings, the general expression of the effective potentials deduced under the algebra can be brought back to the general Natanzon hypergeometric potentials.     

On the path integrations for the Wood-Saxon and related potentials

It is demonstrated that the radial path integral for the Wood-Saxon potential for s-waves can be solved by using the path integration over the SU (2) manifold. The wavefunctions and the energy spectrum are obtained. It is shown that the path integrals for the Rosen-Morse and the Hulthén potentials are also solvable in a similar way.     

Asymptotic iteration approach to supersymmetric bistable potentials

We examine quasi exactly solvable bistable potentials and their supersymmetric partners within the framework of the asymptotic iteration method (AIM). It is shown that the AIM produces excellent approximate spectra and that some×it is found to be more useful to use the partner potential for computation. We also discuss the direct application of the AIM to the Fokker-Planck equation.     

Higher Order Generation of Exactly Solvable Supersymmetric Systems

Starting from an exactly solvable potential, itis shown that it is always possible to construct aninfinite set of higher order generation supersymmetricsystems which are also exactly solvable. A method of deriving these potentials is presented andthe corresponding eigenfunctions and eigenspectra arediscussed. The theory is illustrated with an example inwhich the first- and second-order generation systems are considered.     

Exact Solution of Klein--Gordon Equation by Asymptotic Iteration Method

Using the asymptotic iteration method （AIM） we obtain the spectrum of the Klein-Gordon equation for some choices of scalar and vector potentials. In particular, it is shown that the AIM exactly reproduces the spectrum of some solvable potentials.     

The effect of asymmetry on resonant tunneling in one dimension

This work considers the effect of asymmetric one-dimensional multibarrier potentials on resonant tunneling. It is shown, by using the properties of the propagator of the system, that this effect may lead to novel resonance phenomena which affect the decay properties of the quasistationary states of the system. As a consequence it is shown that in general the full width at half maximum of the transmission peak at resonance does not provide the corresponding lifetime as occurs in the symmetric case. The above considerations are illustrated by a simple analytical solvable model.     

Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schrödinger Hamiltonians

Some exactly solvable potentials in the position dependent mass background are generated whose bound states are given in terms of Laguerre- or Jacobi-type X₁ exceptional orthogonal polynomials. These potentials are shown to be shape invariant and isospectral to the potentials whose bound state solutions involve classical Laguerre or Jacobi polynomials.     

Periodic Quasi-Exactly Solvable Models

Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lamé potential, are shown to emerge naturally in the quantum Hamilton–Jacobi (QHJ) approach. We study the singularity structure of the quantum momentum function, which yields the band-edge eigenvalues and eigenfunctions and compare it with the solvable and quasi-exactly solvable non-periodic potentials, as well as the periodic ones.     

Quantum Deformation of the Two-Dimensional Hydrogen Atom in a Magnetic Field

The deformed Schrodinger equation for thetwo-dimensional hydrogen atom in a homogeneous magneticfield is obtained. It is found that the deformedpotential belongs to a new set of quasi-exactly solvable potentials.     

Equivalence of integral and differential methods of generating accurately solvable potentials of the steady Schrödinger equation

It is proven that the well-known nonlocal (i.e., based on integral transformations) methods of generating accurately solvable potentials of the one-dimensional steady Schrödinger equation are equivalent to multiple use of the local (i.e., based on a differential transformation) method known as the Darboux transformation. New accurately solvable potentials with a hydrogen-like spectrum are obtained, and several functions of the lowest states of the discrete spectrum are presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 95–102, March, 1995.     

Solutions of nonrelativistic Schrödinger equation from relativistic Klein-Gordon equation

The relativistic one-dimensional Klein-Gordon equation can be exactly solved for a certain class of potentials. But the nonrelativistic Schrödinger equation is not necessarily solvable for the same potentials. It may be possible to obtain approximate solutions for the inexact nonrelativistic potential from the relativistic exact solutions by systematically removing relativistic portion. We search for the possibility with the harmonic oscillator potential and the Coulomb potential, both of which can be exactly solvable nonrelativistically and relativistically. Though a rigorous algebraic approach is not deduced yet, it is found that the relativistic exact solutions can be a good starting point for obtaining the nonrelativistic solutions.     

Scattering amplitudes for multi-indexed extensions of solvable potentials

New solvable one-dimensional quantum mechanical scattering problems are presented. They are obtained from known solvable potentials by multiple Darboux transformations in terms of virtual and pseudo virtual wavefunctions. The same method applied to confining potentials, e.g.  Pöschl–Teller and the radial oscillator potentials, has generated the multi-indexed Jacobi and Laguerre polynomials. Simple multi-indexed formulas are derived for the transmission and reflection amplitudes of several solvable potentials.     

Quantum tunneling: A general study in multi-dimensional potential barriers with and without dissipative coupling

Quantum tunneling is formulated in terms of the time evolution of a localized state and thus shown to be dependent upon the eigenspectrum of the system Hamiltonian. A number of exactly solvable models with local and non-local double-well potentials are discussed, and it is shown how, for local potentials, other solvable models can be generated by using Gelfand-Levitan and Darboux transformations. Tunneling in multi-dimensional potential barriers is investigated under semi-classical approximation by developing the method of asymptotic expansion of the wave function for large quantum numbers and the WKB approximation for separable systems. General expressions for the imaginary-time tunneling trajectory are obtained in both methods and specific applications are discussed. Approximation schemes for non-separable systems are also presented. A general study of dissipative multi-dimensional tunneling is carried out by using the Gisin equation, the Schrödinger-Langevin equation and the complex potential model. It is shown that, in general, different models of dissipation are not equivalent in the tunneling context. Using these models one can show (a) the existence of critical damping beyond which no tunneling can occur, (b) that tunneling trajectories are dependent on the damping constant and (c) that dissipation may stabilize the excited state rather than the ground state. Finally the tunneling time delay in one-dimensional systems for undamped and for dissipative systems is formulated in terms of the phase shift, and this has been used to show that the effect of damping on the time delay is ignorable.     

Dynamics of localized states in N = 4 SUSY QM

A consistent approach is offered for investigating the temporal dynamics of localized states. It is based on exactly solvable quantum mechanical models with multi-well potentials and the associate propagators. The Hamiltonian states with multi-well potentials form an adequate basis for expanding wave packets (WP) of various types and degrees of localization. Special features of WP tunneling have been studied with due regard for all Hamiltonian states with symmetric and asymmetric potentials.     

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.     

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.     

两个严格可解双原子分子势函数的散射态解与转动修正

文献［１］中提出了一个新的势函数，其Ｓｃｈｒｄｉｎｇｅｒ方程严格可解。同时提出一个新的变量变换关系，用超几何级数严格求解了双曲型ＰｓｃｈｌＴｅｌｅｒ分子势Ｓｃｈｒｄｉｎｇｅｒ方程的束缚态。文内进一步讨论这两个势函数Ｓｃｈｒｄｉｎｇｅｒ方程散射态的严格解，并求出了Ｓ波的散射相移。文献中关于修正ＰｓｃｈｌＴｅｌｅｒ势及无反射势散射态的结果均作为特例包含在这篇文章更一般的结论之中。此外还用一个简单的方法考虑了转动能修正，对ＨＦ基态转动谱作了具体计算