Minimal Length Uncertainty Relations and New Shape Invariant Models

This paper identifies a new class of shape invariant models. These models are based on extensions of conventional quantum mechanics that satisfy a string-motivated minimal length uncertainty relation. An important feature of our construction is the pairing of operators that are not adjoints of each other. The results in this paper thus show the broader applicability of shape invariance to exactly solvable systems.Comment: 11 pages, no figure

Quasi Exactly Solvable Difference Equations

Several explicit examples of quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/sin^2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.Comment: LaTeX with amsfonts, no figure, 17 pages, a few typos corrected, a reference renewed, 3/2 pages comments on hermiticity adde

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. B679 (2009) 414-417) 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-P\"oschl-Teller potential by a degree \ell (\ell=1,2,...) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3\ell involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities.Comment: 13 page

Conditions for complex spectra in a class of PT symmetric potentials

We study a wide class of solvable PT symmetric potentials in order to identify conditions under which these potentials have regular solutions with complex energy. Besides confirming previous findings for two potentials, most of our results are new. We demonstrate that the occurrence of conjugate energy pairs is a natural phenomenon for these potentials. We demonstrate that the present method can readily be extended to further potential classes.Comment: 13 page

Self-isospectrality, mirror symmetry, and exotic nonlinear supersymmetry

We study supersymmetry of a self-isospectral one-gap Poschl-Teller system in the light of a mirror symmetry that is based on spatial and shift reflections. The revealed exotic, partially broken nonlinear supersymmetry admits seven alternatives for a grading operator. One of its local, first order supercharges may be identified as a Hamiltonian of an associated one-gap, non-periodic Bogoliubov-de Gennes system. The latter possesses a nonlinear supersymmetric structure, in which any of the three non-local generators of a Clifford algebra may be chosen as the grading operator. We find that the supersymmetry generators for the both systems are the Darboux-dressed integrals of a free spin-1/2 particle in the Schrodinger picture, or of a free massive Dirac particle. Nonlocal Foldy- Wouthuysen transformations are shown to be involved in the supersymmetric structure.Comment: 20 pages, comment added. Published versio

Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials

We show that the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems with the trigonometric potential are given by the zeros of the Askey-Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of "discrete" quantum mechanical systems with a q-shift type kinetic term, is shape invariant and the eigenfunctions are the Askey-Wilson polynomials. This is an extension of our previous study [1,2], which established the "discrete analogue" of the well-known fact; The equilibrium positions of the Calogero systems are described by the Hermite and Laguerre polynomials, whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the Hermite and Laguerre polynomials.Comment: 14 pages, 1 figure. The outline of derivation of the result in section 2 is adde

Unified treatment of the Coulomb and harmonic oscillator potentials in DD dimensions

Quantum mechanical models and practical calculations often rely on some exactly solvable models like the Coulomb and the harmonic oscillator potentials. The $D$ dimensional generalized Coulomb potential contains these potentials as limiting cases, thus it establishes a continuous link between the Coulomb and harmonic oscillator potentials in various dimensions. We present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We define a Hilbert space basis, the generalized Coulomb-Sturmian basis, and calculate the Green's operator on this basis and also present an SU(1,1) algebra associated with it. We formulate the problem for the one-dimensional case too, and point out that the complications arising due to the singularity of the one-dimensional Coulomb problem can be avoided with the use of the generalized Coulomb potential.Comment: 18 pages, 3 ps figures, revte

Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm

Several kinds of q-orthogonal polynomials with |q|=1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems. Their orthogonality weight functions consist of quantum dilogarithm functions, which are a natural extension of the Euler gamma functions and the q-gamma functions (q-shifted factorials). The dimensions of the orthogonal spaces are finite. These q-orthogonal polynomials are expressed in terms of the Askey-Wilson polynomials and their certain limit forms.Comment: 37 pages. Comments and references added. To appear in J.Math.Phy

Orthogonal Polynomials from Hermitian Matrices

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger equations. The hermitian matrices (factorisable Hamiltonians) are real symmetric tri-diagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalisation measures and the normalisation constants etc. are determined explicitly.Comment: 53 pages, no figures. Several sentences and a reference are added. To be published in J. Math. Phy

New Two-Dimensional Quantum Models with Shape Invariance

Two-dimensional quantum models which obey the property of shape invariance are built in the framework of polynomial two-dimensional SUSY Quantum Mechanics. They are obtained using the expressions for known one-dimensional shape invariant potentials. The constructed Hamiltonians are integrable with symmetry operators of fourth order in momenta, and they are not amenable to the conventional separation of variables.Comment: 16 p.p., a few new references adde