Solvable Potentials Research Articles

Generating QES potentials supporting zero energy normalizable states for an extended class of truncated Calogero Sutherland model

Motivated by recent interest in the search for generating potentials for which the underlying Schrödinger equation is solvable, we report in the recent work several situations when a zero-energy state becomes bound depending on certain restrictions on the coupling constants that define the potential. In this regard, we present evidence of the existence of regular zero-energy normalizable solutions for a system of quasi-exactly solvable (QES) potentials that correspond to the rationally extended many-body truncated Calogero–Sutherland (TCS) model. Our procedure is based upon the use of the standard potential group approach with an underlying so(2,1) structure that utilizes a point canonical transformation with three distinct types of potentials emerging having the same eigenvalues while their common properties are subjected to the evaluation of the relevant wave functions. These cases are treated individually by suitably restricting the coupling parameters.

Supersymmetric Expansion Algorithm and Complete Analytical Solution for the Hulthén and Anharmonic Potentials

Abstract An algorithm for providing analytical solutions to Schrödinger’s equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the supersymmetric quantum mechanics as extended toward non shape invariant potentials. The complete solution to a given Hamiltonian H0 is obtained from the nodeless states of the Hamiltonian H0 and of a set of supersymmetric partners H1, H2, …, Hr. The nodeless states (dubbed “edge” states) are unique and in general can be ground or excited states. They are solved using the logarithmic expansion which yields an infinite system of coupled first order hierarchical differential equations, converted later into algebraic equations with recurrence relations which can be solved order by order. We formulate the aforementioned scheme, termed to as “Supersymmetric Expansion Algorithm” step by step and apply it to obtain for the first time the complete analytical solutions of the three dimensional Hulthén–, and the one-dimensional anharmonic oscillator potentials.

Potentials on the conformally compactified Minkowski spacetime and their application to quark deconfinement

In this paper, we study a class of conformal metric deformations in the quasi-radial coordinate parametrizing the three-sphere in the conformally compactified Minkowski spacetime [Formula: see text]. Prior to reduction of the associated Laplace–Beltrami operators to a Schrödinger form, a corresponding class of exactly solvable potentials (each one containing a scalar and a gradient term) is found. In particular, the scalar piece of these potentials can be exactly or quasi-exactly solvable, and among them we find the finite range confining trigonometric potentials of Pöschl–Teller, Scarf and Rosen–Morse. As an application of the results developed in the paper, the large compactification radius limit of the interaction described by some of these potentials is studied, and this regime is shown to be relevant to a quantum mechanical quark deconfinement mechanism.

Schrödinger equation as a confluent Heun equation

This article deals with two classes of quasi-exactly solvable (QES) trigonometric potentials for which the one-dimensional Schrödinger equation reduces to a confluent Heun equation (CHE) where the independent variable takes only finite values. Power series for the CHE are used to get polynomial and nonpolynomial eigenfunctions. Polynomials occur only for special sets of parameters and characterize the quasi-exact solvability. Nonpolynomial solutions occur for all admissible values of the parameters (even for values which give polynomials), and are bounded and convergent in the entire range of the independent variable. Moreover, throughout the article we examine other QES trigonometric and hyperbolic potentials. In all cases, for a polynomial solution there is a convergent nonpolynomial solution.

A new conjecture for obtaining the energy spectra of the wave equations under solvable potentials

A new conjecture for obtaining the energy spectra of the wave equations under solvable potentials

A class of exactly solvable real and complex PT symmetric reflectionless potentials

We consider the question of the number of exactly solvable complex but PT-invariant reflectionless potentials with N bound states. By carefully considering the Xm rationally extended reflectionless potentials, we argue that the total number of exactly solvable complex PT-invariant reflectionless potentials are 2[(2N − 1)m + N].

Quasi-exactly solvable potentials in Wigner-Dunkl quantum mechanics

It is shown that the Dunkl harmonic oscillator on the line can be generalized to a quasi-exactly solvable one, which is an anharmonic oscillator with known eigenstates for any . It is also proved that the Hamiltonian of the latter can also be rewritten in a simpler way in terms of an extended Dunkl derivative. Furthermore, the Dunkl isotropic oscillator and Dunkl Coulomb potentials in the plane are generalized to quasi-exactly solvable ones. In the former case, potentials with known eigenstates are obtained, whereas, in the latter, sets of potentials associated with a given energy are derived.

Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials.

Rationally extended harmonic oscillator potential, isospectral family and the uncertainty relations

We consider the rationally extended harmonic oscillator potential which is isospectral to the conventional one and whose solutions are associated with the exceptional, Xm- Hermite polynomials and discuss its various important properties for different even codimension of m. The uncertainty relations are obtained for different m and it is shown that for the ground state, the uncertainty increases as m increases. A one parameter (λ) family of exactly solvable isospectral potential corresponding to this extended harmonic oscillator potential is obtained. Special cases corresponding to the λ=0 and λ=−1, which give the Pursey and the Abraham-Moses potentials respectively, are discussed. The uncertainty relations for the entire isospectral family of potentials for different m and λ are also calculated.

Unified Supersymmetric Description of Shape-Invariant Potentials within and beyond the Natanzon Class

The transformations of supersymmetric quantum mechanics are discussed within a formalism that employs a six-parameter function, from which the superpotential and the supersymmetric partner potentials V−(x) and V+(x) are constructed in a general form. By specific choice of the parameters, V−(x) and V+(x) are matched with the general form of PI class potentials and their rationally extended versions. The choice of the parameters also determines which of the four possible SUSY transformations Ti, i=1,…4 is in effect. After this general discussion, the formulae are specified to the three members of this potential class, the Scarf I, Scarf II and generalized Pöschl–Teller potentials. Due to the different domains of definition and their consequences on the boundary conditions, the results turn out to be rather diverse for the three potentials, while the mathematical formalism and the network of the potentials interconnected by the SUSYQM transformations still remains common to a large extent. The general framework allows a unified and consistent interpretation of earlier isolated findings. It also helps to connect the results to further potential classes and to place them into a more general context within the zoo of exactly solvable potentials.

Supersymmetric quantum potentials analogs of classical electrostatic fields

A relation between classical electrostatic fields and Shrödinger-like Hamiltonians is evidenced. Hence, supersymmetric quantum potentials analogous to classical electrostatic fields can be constructed. Proposing an ansatz for the electrostatic potential as the natural logarithm of a nodeless function, it is demonstrated that the electrostatic fields fulfil the Bernoulli equation associated to a second-order confluent supersymmetric transformation. By using the so-called confluent algorithm, it is possible, given a charge density, to find the corresponding electrostatic field as well as the supersymmetric potentials. Furthermore, the associated charge density and the electrostatic field profile of Schrödinger-like solvable potentials can be determined.

Conditionally exactly solvable Dirac potential, including x 1/3 pseudoscalar interaction

We study the solution of the 1D Dirac equation for a pseudoscalar potential, which includes an interaction term proportional to and a term with a fixed strength. This is a conditionally exactly solvable singular potential, symmetric with respect to the origin. A feature of the potential is that the effective potential for the Schrödinger equation, to which the Dirac equation can be reduced, splits into two known potentials: the first Stillinger potential and the second Exton potential for the positive and negative semiaxes, respectively. We present the solution and analyze the energy spectrum and wave functions of the bound states. Our results reveal notable differences between the Schrödinger and Dirac behaviors.

Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient

This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at 0 and ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty$$\\end{document} of the Laurent polynomial, is seen as an affine algebraic variety. We prove that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies, such as the doubly confluent and biconfluent Heun equations, and to the theory of algebraically solvable potentials of Shrödinger equations. Also, as an auxiliary tool, we improve a previously known criterium for a second order linear differential equations to admit a polynomial solution.

The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics

Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.

Comment on “many-body quantum system in the presence of solvable potential”

Comment on “many-body quantum system in the presence of solvable potential”

Quasi-exactly solvable extensions of the Kepler–Coulomb potential on the sphere

We consider a family of extensions of the Kepler–Coulomb potential on a d-dimensional sphere and analyze it in a deformed supersymmetric framework, wherein the starting potential is known to exhibit a deformed shape invariance property. We show that the members of the extended family are also endowed with such a property, provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build quasi-exactly solvable potentials with known ground and first-excited states. Some explicit results are obtained for the first three members of the family. We then use a generating function method, wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function W+(r) [and its accompanying function W−(r)]. From the results obtained for the latter for the first three family members, we propose some formulas for W±(r) valid for the mth family member, depending on m+1 constants a0, a1, …, am. Such constants satisfy a system of m+1 linear equations. Solving the latter allows us to extend the results up to the seventh family member and then to formulate a conjecture giving the general structure of the ai constants in terms of the parameters of the problem.

Potentials from the Polynomial Solutions of the Confluent Heun Equation

Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function Hc(p,β,γ,δ,σ;z). One of these recovers the generalized Laguerre polynomials LN(α), and another one the rationally extended X1 type Laguerre polynomials L^N(α). The two remaining solutions represent previously unknown polynomials that do not form an orthogonal set and exhibit features characteristic of semi-classical orthogonal polynomials. A standard method of generating exactly solvable potentials in the one-dimensional Schrödinger equation is applied to the CHE, and all known potentials with solutions expressed in terms of the generalized Laguerre polynomials within, or outside the Natanzon confluent potential class, are recovered. It is also found that the potentials generated from the two new polynomial systems necessarily depend on the N quantum number. General considerations on the application of the Heun type differential differential equations within the present framework are also discussed.

Quasi-exact treatment of non-relativistic generalized hyperbolic potentials

The solution of the Schrödinger equation for the two quasi-exactly solvable potentials is presented using the Lie algebra approach. It is shown that all models give rise to the same basic differential equation which is quasi-exactly solvable. The eigenvalues, eigenfunctions and the allowed potential parameters are given for each of the two models in terms of the roots of a set of algebraic quasi-exact solvable methods.

Jacobi partial waves for a set of 3D noncentral rational scatterers

The common tool of choice for basis expansions for the scattering problem with 3D quantum systems remains the spherical harmonics as eigenfunctions of the Laplace–Beltrami operator on the sphere, with approximations for deviations made around the usually dominant s-wave spherically symmetric state. However, with the growing number of technologically accessible nonspherically symmetric geometries of cold atomic and molecular systems, there is a need to explore as orthonormal bases for partial wave analysis the larger class of weighted Jacobi polynomials, subsuming the spherical harmonics. In particular, the lowest angular state for this bigger class is a toroid instead of a spherical s-orbital. This allows analytic treatment of a wider array of rational angular-dependent potentials which can describe rings and systems with topological constraints such as monopoles. Here, we present exact analytic solutions for the quantum scattering problem by partial wave analysis using the weighted Jacobi polynomials as an expanded basis. We obtain the scattering amplitude, differential and total cross-sections for exactly solvable 3D potentials included in the Smorodinsky-Winternitz noncentral systems with dynamical symmetries. Moreover, this procedure also solves the quantum scattering problem from a novel series of rational trigonometric forms of anisotropic potentials including double ring-shaped configurations.

A new class of solvable two-dimensional scalar potentials for graphene

In the present paper, a systematic approach is presented for solution of two-dimensional massless Dirac equation with external electrostatic potential applied. This approach is based on the new - asymmetric - form of SUSY-like intertwining relations. It allows to build a wide variety of pairs of SUSY-partner external scalar potentials. If one of them is simple enough to be solvable, its partner is also solvable although it may have a non-trivial dependency on both coordinates. Physically, this kind of problems is related to the description of graphene and some other materials with external potential. Solvability obtained by means of asymmetric form of SUSY intertwining relations allows to extend the class of analytically solvable two-dimensional models.