Exactly solvable associated Lamé potentials and supersymmetric transformations

A systematic procedure to derive exact solutions of the associated Lamé equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are used to generate new exactly solvable potentials; some of them exhibit an interesting property of periodicity defects.     

Exactly complete solutions of the Schrödinger equation with a spherically harmonic oscillatory ring-shaped potential

A spherically harmonic oscillatory ring-shaped potential is proposed and its exactly complete solutions are presented by the Nikiforov-Uvarov method. The effect of the angle-dependent part on the radial solutions is discussed.     

The many-body problem with an energy-dependent confining potential

We consider systems of N bosons bound by two-body harmonic interactions, whose frequency depends on the total energy of the system. Such energy dependent confining interactions between the bosons yield remarkable properties of the many-body system. As the quantum numbers increase, the total energy cannot exceed a saturation energy, which is independent of the number of particles N. Moreover, the ground state energy increases with N. As a result, the density of states tends rapidly to infinity as N and/or the quantum numbers increase.     

A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

The quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx²)⁻¹ and with a λ-dependent non-polynomial rational potential. This λ-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for λ → 0 all the characteristics of the linear oscillator are recovered. First, the λ-dependent Schrödinger equation is exactly solved as a Sturm-Liouville problem, and the λ-dependent eigenenergies and eigenfunctions are obtained for both λ > 0 and λ < 0. The λ-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as λ-deformations of the standard Hermite polynomials. In the second part, the λ-dependent Schrödinger equation is solved by using the Schrödinger factorization method, the theory of intertwined Hamiltonians, and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a λ-dependent Rodrigues formula, a generating function and λ-dependent recursion relations between polynomials of different orders.     

Analytical approximation to the solution of the Dirac equation with the Eckart potential including the spin-orbit coupling term

By using the supersymmetric WKB approximation approach and the functional analysis method, we solve approximately the Dirac equation with the Eckart potential for the arbitrary spin-orbit quantum number κ. The bound state energy eigenvalues and the associated two-component spinors of the Dirac particles are obtained approximately.     

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.     

Two hierarchies of integrable lattice equations associated with a discrete matrix spectral problem

Two hierarchies of nonlinear integrable positive and negative lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws of the positive hierarchy, then, the integrable coupling systems of the positive hierarchy are derived from enlarging Lax pair.     

Exact solution of the Dirac equation with the Mie-type potential under the pseudospin and spin symmetry limit

We investigate the exact solution of the Dirac equation for the Mie-type potentials under the conditions of pseudospin and spin symmetry limits. The bound state energy equations and the corresponding two-component spinor wave functions of the Dirac particles for the Mie-type potentials with pseudospin and spin symmetry are obtained. We use the asymptotic iteration method in the calculations. Closed forms of the energy eigenvalues are obtained for any spin-orbit coupling term κ. We also investigate the energy eigenvalues of the Dirac particles for the well-known Kratzer-Fues and modified Kratzer potentials which are Mie-type potentials.     

An extended class of L-series solutions of the wave equation

We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such includes the discrete (for bound states) as well as the continuous (for scattering states) spectrum of the Hamiltonian. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, include problems in one and three dimensions.     

Wave equations with energy-dependent potentials

We study wave equations with energy-dependent potentials. Simple analytical models are found useful to illustrate difficulties encountered with the calculation and interpretation of observables. A formal analysis shows under which conditions such equations can be handled as evolution equation of quantum theory with an energy-dependent potential. Once these conditions are met, such theory can be transformed into ordinary quantum theory. This work was supported by the agreement between IN2P3 and ASCR (collaboration no. 97-13) and by the Grant Agency of ASCR (J.M., grant No.A1048305).     

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.     

Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey–Wilson polynomials in terms of a degree ?   (?=1,2,…$?=1,2,\dots$) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree ??1$??1$ and thus not constrained by any generalisation of Bochner's theorem.     

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.     

Existence conditions and spreading properties of extreme entropy D-dimensional distributions

The extremization of the information-theoretic measures (Fisher information, Shannon entropy, Tsallis entropy), which complementary describe the spreading of the physical states of natural systems, gives rise to fundamental equations of motion and/or conservation laws. At times, the associated extreme entropy distributions are known for some given constraints, usually moments or radial expectation values. In this work, first we give the existence conditions of the maxent probability distributions in a D-dimensional scenario where two moments (not necessarily of consecutive order) are known. Then we find general relations which involve four elements (the extremized entropy, the other two information-theoretic measures and the variance of the extremum density) in scenarios with different dimensionalities and moment constraints.     

An Improvement of the Asymptotic Iteration Method for Exactly Solvable Eigenvalue Problems

We derive a formula that simplifies the original asymptotic iteration method formulation to find the energy eigenvalues for the analytically solvable cases. We then show that there is a connection between the asymptotic iteration and the Nikiforo-Uvarov methods, which both solve the second order linear ordinary differential equations analytically.     

Ground State Eigenfunction of Spheroidal Wave Functions

We study the spin-weighted spheroidal wave functions in the case of s = m = 0. Their eigenvalue problem is investigated by the perturbation method in supersymmetric quantum mechanics. In the first three terms of parameter a = a^2 w^2, the ground eigenvalue and eigenfunction are obtained. The obtained ground eigenfunction is elegantly in dosed forms. These results are new and very useful for the application of the spheroidal wave functions.     

Exactly solvable models in atomic and molecular physics

We construct integrable generalized models in a systematic way exploring different representations of the gl(N)$\mathit{gl}(N)$ algebra. The models are then interpreted in the context of atomic and molecular physics, most of them related to different types of Bose–Einstein condensates. The spectrum of the models is given through the analytical Bethe ansatz method. We further extend these results to the case of the superalgebra gl(M|N)$\mathit{gl}(M|N)$, providing in this way models which also include fermions.     

Schrödinger equation with delta potential in superspace

A superspace version of the Schrödinger equation with a delta potential is studied using Fourier analysis. An explicit expression for the energy of the single bound state is found as a function of the super-dimension M in case M is smaller than or equal to 1. In the case when there is one commuting and 2n anti-commuting variables also the wave function is given explicitly.