152 research outputs found

### 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 $D$ 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

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