170 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
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
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
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
Solvable rational extensions of the Morse and Kepler-Coulomb potentials
We show that it is possible to generate an infinite set of solvable rational
extensions from every exceptional first category translationally shape
invariant potential. This is made by using Darboux-B\"acklund transformations
based on unphysical regular Riccati-Schr\"odinger functions which are obtained
from specific symmetries associated to the considered family of potentials
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
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
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