Darboux transformation and elementary exact solutions of the Schrödinger equation

Darboux transformation is applied to three classical potentials, namely the harmonic oscillator, effective Coulomb and Morse potentials to generate exactly solvable potentials of elementary form. For every potential, the isospectral families of potentials are constructed. For almost all potentials, a set of normalized discrete spectrum wave functions is given.     

Exactly solvable two-dimensional stationary Schrödinger operators obtained by the nonlocal Darboux transformation

The Fokker–Planck equation associated with the two-dimensional stationary Schrödinger equation has the conservation law form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schrödinger equation that provides the nonlocal Darboux transformation for the Schrödinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two-dimensional stationary Schrödinger equations. The examples of exactly solvable two-dimensional stationary Schrödinger operators with smooth potentials decaying at infinity are obtained.     

On askey-wilson algebra

The ladder representations of Askey-Wilson algebra (AWA) is investigated and the over-lap problem is discussed. The full class of exactly solvable potentials for one-dimensional ordinary Schrödinger equation with the AWA as an algebra of dynamical symmetry is found for creation-annihilation operators of third order. The generating spectrum algebra for latticeSchrödinger equation as AWA is examined. All exactly solvable potentials with this dynamical symmetry are found. Some generalizations of obtained results are discussed.     

Generation of Exactly Solvable Non-Hermitian Potentials with Real Energies

Series of exactly solvable non-trivial complex potentials (possessing real spectra) are generated by applying the Darboux transformation to the excited eigenstates of a non-Hermitian potential V(x). This method yields an infinite number of non-trivial partner potentials, defined over the whole real line, whose spectra are nearly exactly identical to the original potential.     

The Darboux Transform for the One-Dimensional Stationary Dirac Equation

Based on the method oftransformation operators, the Darboux transformation operator has been constructed for the one-dimensional stationary Dirac equation. The properties of this operator have been studied. As an application, exactly solvable transparent potentials and potentials with the spectrum of a relativistic harmonic oscillator have been obtained.     

A class of exactly solvable Schrödinger equation with moving boundary condition

Using first and second order supersymmetry formalism we obtain a class of exactly solvable potentials subject to moving boundary condition.     

Nonrelativistic Green's Function for Systems with Position-Dependent Mass

Given a spatially dependent mass, we obtain the 2-point Green's function for exactly solvable nonrelativistic problems. This is accomplished by mapping the wave equation for these systems into well-known exactly solvable Schrödinger equations with constant mass using point canonical transformation. The one-dimensional oscillator class is considered and examples are given for several mass distributions.     

Exact solution to a class of generalized Kitaev spin-1/2 models in arbitrary dimensions

We construct a class of exactly solvable generalized Kitaev spin-1/2 models in arbitrary dimensions, which is beyond the category of quantum compass models. The Jordan-Wigner transformation is employed to prove the exact solvability. An exactly solvable quantum spin-1/2 model can be mapped to a gas of free Majorana fermions coupled to static Z2 gauge fields. We classify these exactly solvable models according to their parent models. Any model belonging to this class can be generated by one of the parent models. For illustration, a two dimensional(2D) tetragon-octagon model and a three dimensional(3D) xy bond model are studied.     

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.     

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     

SUSY formalism for the symmetric double well potential

Using first- and second-order supersymmetric Darboüx formalism and starting with symmetric double well potential barrier we have obtained a class of exactly solvable potentials subject to moving boundary condition. The eigenstates are also obtained by the same technique.     

The conditionally solvable natanzon potentials

It is shown how to derive the recently introduced conditionally solvable Natanzon potentials in the framework of the Schrödinger’s formulation of quantum mechanics. Their relation to the exactly solvable potentials is considered. It is shown that in this way the variety of conditionally solvable potentials can be obtained. The relationship between the Natanzon potentials and shape invariant ones is derived.     

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：     

两个严格可解的双原子分子势函数

提出了一个新的势函数，其Ｓｃｈｒ?ｄｉｎｇｅｒ方程严格可解．同时提出一个新的变量变换关系，严格求解了双曲型Ｐ?ｓｃｈｌ-Ｔｅｌｅｒ分子势Ｓｃｈｒ?ｄｉｎｇｅｒ方程的束缚态．得出的能级公式形式封闭、简单，归一化波函数用超几何级数表达．文献中关于Ｈｕｌｔｈéｎ势和修正Ｐ?ｓｃｈｌ-Ｔｅｌｅｒ势及无反射势的结果都作为特例包含在本文更一般的结论之中．最后，还讨论了本文势函数对双原子分子的实际应用． 关键词：     

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.     

Complex oscillator and Painlevé IV equation

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second-order supersymmetric transformations will be used to obtain new exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians turn out to be ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlevé IV equation is achieved, thus giving place to new solutions for the Painlevé IV equation.     

Generation of exact analytic bound state solutions from solvable non-powerlaw potentials by a transformation method

A transformation method has been applied to the exactly solvable Hulthen problem to generate a hierarchy of exactly solved quantum systems in any chosen dimension. The generated quantum systems are, in general, energy-dependent with a single normalized eigenfunction, as the Hulthen potential is a non-powerlaw potential. A method has been devised to convert a subset of the generated quantum systems with energy-dependent potentials to a single normal system with an energy-independent potential that behaves like a potential qualitatively similar to the Poschl-Teller potential. A second-order application of the transformation method on the Hulthen system produces another Sturmian quantum system and a different method is given to regroup them into a normal quantum system which resembles the Morse potential. Existence of normalizable eigenfunctions for these systems are found to be dependent on the local and asymptotic behaviour of the transformation function. Received 30 August 2000 and Received in final form 16 March 2001     

Simple unified derivation and solution of Coulomb, Eckart and Rosen-Morse potentials in prepotential approach

The four exactly solvable models related to non-sinusoidal coordinates, namely, the Coulomb, Eckart, Rosen-Morse type I and II models are normally being treated separately, despite the similarity of the functional forms of the potentials, their eigenvalues and eigenfunctions. Based on an extension of the prepotential approach to exactly and quasi-exactly solvable models proposed previously, we show how these models can be derived and solved in a simple and unified way.     

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.     

Coherent Schwinger interaction from Darboux transformation

The exactly solvable scalar-tensor potential of the four-component Dirac equation has been obtained by the Darboux transformation method. The constructed potential has been interpreted in terms of nucleon-nuclear and Schwinger interactions of neutral particles with lattice sites during their channeling in the nonmagnetic crystal. The family of exactly solvable interaction Hamiltonians of a Schwinger type is obtained by means of the Darboux transformation chain. The analytic structure of the Lyapunov function of periodic continuation for each of the Hamiltonians of the family is considered.