276 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
Asymptotic Iteration Method Solutions to the Relativistic Duffin-Kemmer-Petiau Equation
A simple exact analytical solution of the relativistic Duffin-Kemmer-Petiau
equation within the framework of the asymptotic iteration method is presented.
Exact bound state energy eigenvalues and corresponding eigenfunctions are
determined for the relativistic harmonic oscillator as well as the Coulomb
potentials. As a non-trivial example, the anharmonic oscillator is solved and
the energy eigenvalues are obtained within the perturbation theory using the
asymptotic iteration method.Comment: 17 pages written with LaTeX Revtex4. accepted for publication in
Journal of Mathematical Physic
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
Duality and Central Charges in Supersymmetric Quantum Mechanics
We identify a class of point-particle models that exhibit a target-space
duality. This duality arises from a construction based on supersymmetric
quantum mechanics with a non-vanishing central charge. Motivated by analogies
to string theory, we are led to speculate regarding mechanisms for restricting
the background geometry.Comment: 10 pages, 1 figure, late
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
New exactly solvable relativistic models with anomalous interaction
A special class of Dirac-Pauli equations with time-like vector potentials of
external field is investigated. A new exactly solvable relativistic model
describing anomalous interaction of a neutral Dirac fermion with a
cylindrically symmetric external e.m. field is presented. The related external
field is a superposition of the electric field generated by a charged infinite
filament and the magnetic field generated by a straight line current. In
non-relativistic approximation the considered model is reduced to the
integrable Pron'ko-Stroganov model.Comment: 20 pages, discussion of the possibility to test the model
experimentally id added as an Appendix, typos are correcte
Unified treatment of the Coulomb and harmonic oscillator potentials in dimensions
Quantum mechanical models and practical calculations often rely on some
exactly solvable models like the Coulomb and the harmonic oscillator
potentials. The 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
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