Solvable Potentials Research Articles

New quasi-exactly solvable periodic potentials

We present two new classes of quasi-exactly solvable periodic potentials. It is shown that their solutions are given in terms of a known special function. When the parameters in the two periodic potentials satisfy a certain condition, some of the Bloch waves and associated energies can be found exactly and in closed form at either the center or the edge of the first Brillouin zone. Two fundamental sets of solutions have been constructed to calculate all the Bloch waves and energy bands.

Generalized factorization and isospectral potentials

It is shown that the method of using excited-state wave functions in constructing exactly solvable potentials as suggested by Berger and Ussembayev [Phys. Rev. A 82, 022121 (2010)] can be generalized further. Using this method, exactly solvable potentials related to the harmonic oscillator and some other potentials are obtained. The nonlinear symmetry algebras underlying the harmonic-oscillator models are also obtained explicitly.

Two-dimensional Darboux transformations for non-separable angular equations and solvable non-central potentials

We construct a two-dimensional Darboux transformation for the angular equation that arises from variable separation of a Schrodinger equation in spherical coordinates. Since we do not assume the angular equation to be further separable, our Darboux transformation allows for the systematic generation of solvable non-central potentials that are not accessible through the usual separation of variables.

Hypergeometric type operators and their supersymmetric partners

The generalization of the factorization method performed by Mielnik [J. Math. Phys. 25, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to hypergeometric type operators. It is based on some solvable Riccati equations and leads to a unitary description of the quantum systems exactly solvable in terms of orthogonal polynomials or associated special functions.

Dirac Equation and Ground State of Solvable Potentials: Supersymmetry Method

The supersymmetry in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We show that some of the superpotentials related to ground state of the solvable potentials in non-relativistic quantum mechanics can be used for studying of the Dirac equation.

A new family of shape invariantly deformed Darboux–Pöschl–Teller potentials with continuous ℓ

We present a new family of shape invariant potentials which could be called a ‘continuous ℓ version’ of the potentials corresponding to the exceptional (Xℓ) J1 Jacobi polynomials constructed recently by the present authors. In a certain limit, it reduces to a continuous ℓ family of shape invariant potentials related to the exceptional (Xℓ) L1 Laguerre polynomials. The latter was known as one example of the ‘conditionally exactly solvable potentials’ on a half line.

Dirac equation with position-dependent effective mass and solvable potentials in the Schrödinger equation

We study the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the framework of the position-dependent effective mass Dirac equation. The Dirac equation is mapped into the exactly solvable Schrödinger-like equation endowed with position-dependent effective mass that we present a new procedure to solve it. The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function, the imaginary potential can be obtained. The spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials. We also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials.

Information entropy of conditionally exactly solvable potentials

We evaluate Shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials. The Bialynicki–Birula–Mycielski inequality has also been tested for a number of states.

Exact analytic solutions generated from stipulated Morse and trigonometric Scarf potentials

The extended transformation method has been applied to the exactly solvable stipulated Morse potential and trigonometric Scarf potential, to generate a set of exactly solvable quantum systems (QSs) in any chosen dimension. Bound state solutions of the exactly solvable potentials are given. The generated QSs are generally of Sturmian form. We also report a system case-specific regrouping technique to convert a Sturmian QS to a normal QS. A second-order application of the transformation method is given. The normalizability of the generated QSs is generally given in Sturmian form.

The Infinite Square Well with a Singular Perturbation

We study the influence of a singular potential on an infinite square well. Two cases are considered. In the first, the singular potential is centered in the potential well, in the second case it is off-center. We also present a discussion on the self adjointness of the Hamiltonian of some exactly solvable potentials perturbed with singular potentials.

Galoisian approach to integrability of Schrödinger equation

In this paper, we examine the nonrelativistic stationary Schrodinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second-order ordinary linear differential operators, so as to achieve rational function coefficients (“algebrization”), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schrodinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.

Shape Invariant Potentials in Second-Order Supersymmetric Quantum Mechanics

Shape invariance condition in the framework of second-order supersymmetric quantum mechanics is studied. Two classes of solvable shape invariant potentials are consequently constructed, in which the parameters a0 and a1 of partner potentials are related to each other by translation a1 = a0 + ?. In each class, general properties of the obtained shape invariant potentials are systematically investigated. The energy eigenvalues are algebraically determined and the corresponding eigenfunctions are expressed in terms of generalized associated Laguerre polynomials. It is found that these shape invariant potentials are inherently singular, characterized by the 1/x2 singularity at the origin. MSC 2010: 81Q60, 81Q80.

Asymptotic Properties of Solvable $\mathcal{PT}$ -Symmetric Potentials

The asymptotic region of potentials has strong impact on their general properties. This problem is especially interesting for $\mathcal{PT}$ -symmetric potentials, the real and imaginary components of which allow for a wider variety of asymptotic properties than in the case of purely real potentials. We consider exactly solvable potentials defined on an infinite domain and investigate their scattering and bound states with special attention to the boundary conditions determined by the asymptotic regions. The examples include potentials with asymptotically vanishing and non-vanishing real and imaginary potential components (Scarf II, Rosen-Morse II, Coulomb). We also compare the results with the asymptotic properties of some exactly non-solvable $\mathcal{PT}$ -symmetric potentials. These studies might be relevant to the experimental realization of $\mathcal{PT}$ -symmetric systems.

Near-exact supersymmetric partner potentials: Construction and exploitation

Precise supersymmetric (SUSY) partner potentials can be generated only for exactly solvable problems of the stationary Schrodinger equation. This is a severe restriction, as most problems are not amenable to exact solutions. We employ here a linear variational strategy to explicitly construct approximate SUSY partners of a few common, not exactly solvable potentials and subsequently examine their properties to explore the advantages in practical implementation. The efficacy of our proposed scheme is commendable. We demonstrate that, for symmetric potentials, the constructed partners may be so good that the overall recipe has the nicety of generating the whole eigenspectrum by employing only half of the full Hilbert space functions. A similar strategy is shown to work for the odd states too, with proper boundary conditions. Pilot calculations involve a number of low-lying states of some mixed oscillator and double-well potentials. Analysis of the results reveals a few interesting features of the problem of construction of approximate SUSY partners and their practical use. Particularly, we identify places where the operator-level approximations are involved and how far they affect the bounding properties of energies that are obtained as eigenvalues of a matrix diagonalization problem associated with linear variations. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Two-dimensional stationary Schrödinger equation via the ∂¯-dressing method: New exactly solvable potentials, wave functions, and their physical interpretation

The classes of exactly solvable multiline soliton potentials and corresponding wave functions of two-dimensional stationary Schrödinger equation via ∂¯-dressing method are constructed and their physical interpretation is discussed.

THE QUANTIZATION RULE AND MASLOV INDEX

Within extensions of the new quantization rule approach in arbitrary dimensions, the Maslov indices and energy spectra of some exactly solvable potentials are presented. We find that the Maslov index for the harmonic oscillator in three dimensions agrees well with those obtained by other methods.

Generation of new classes of exactly solvable potential from the trigonometric Rosen-Morse potential

Exact analytic solutions of the Schrodinger equation are obtained for classes of newly constructed potentials which are generated from the trigonometric Rosen-Morse potential as the input reference potential via extended transformation method. A set of quantized energy spectra of the bound states and the corresponding wave functions of the generated potentials are obtained. We also focus on to the Romanovski Polynomials which is a family of the real orthogonal polynomials and is required to present exact real analytic solutions of the generated potentials.

Infinitely many shape-invariant potentials and cubic identities of the Laguerre and Jacobi polynomials

We provide analytic proofs for the shape invariance of the recently discovered [Odake and Sasaki, Phys. Lett. B 679, 414 (2009)] two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux–Pöschl–Teller potential by a degree ℓ (ℓ=1,2,…) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3ℓ involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities.

Conditionally exactly solvable potentials and exceptional orthogonal polynomials

It is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials. Some properties of such polynomials, e.g., recurrence relation, ladder operators, differential equations, etc., have been obtained.

Solving a coupled-channels scattering problem by adding confining potentials

A new method to obtain both the wave function and S matrix for coupled-channels scattering problems is presented. The method is characterized by the introduction of a real confining potential acting in the external region, which enables one to solve the problem using virtually only the bound-state solutions. The asymptotic behavior of the continuum state is guaranteed with a single-channel Green's function. The accuracy and simplicity of the method are demonstrated on the two examples of an exactly solvable two-channel potential and the nucleon–nucleon scattering.