604 research outputs found
Deformed Heisenberg algebra and minimal length
A one-dimensional deformed Heisenberg algebra is studied. We
answer the question: For what function of deformation there exists a
nonzero minimal uncertainty in position (minimal length). We also find an
explicit expression for the minimal length in the case of arbitrary function of
deformation.Comment: to be published in JP
Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials
We present four types of infinitely many exactly solvable Fokker-Planck
equations, which are related to the newly discovered exceptional orthogonal
polynomials. They represent the deformed versions of the Rayleigh process and
the Jacobi process.Comment: 17 pages, 4 figure
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
On Dirac theory in the space with deformed Heisenberg algebra. Exact solutions
The Dirac equation has been studied in which the Dirac matrices
\hat{\boldmath\alpha}, \hat\beta have space factors, respectively and
, dependent on the particle's space coordinates. The function deforms
Heisenberg algebra for the coordinates and momenta operators, the function
being treated as a dependence of the particle mass on its position. The
properties of these functions in the transition to the Schr\"odinger equation
are discussed. The exact solution of the Dirac equation for the particle motion
in the Coulomnb field with a linear dependence of the function on the
distance to the force centre and the inverse dependence on for the
function has been found.Comment: 13 page
Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces
We introduce two new families of quasi-exactly solvable (QES) extensions of
the oscillator in a -dimensional constant-curvature space. For the first
three members of each family, we obtain closed-form expressions of the energies
and wavefunctions for some allowed values of the potential parameters using the
Bethe ansatz method. We prove that the first member of each family has a hidden
sl(2,) symmetry and is connected with a QES equation of the first
or second type, respectively. One-dimensional results are also derived from the
-dimensional ones with , thereby getting QES extensions of the
Mathews-Lakshmanan nonlinear oscillator.Comment: 30 pages, 8 figures, published versio
Spectrum generating algebras for position-dependent mass oscillator Schrodinger equations
The interest of quadratic algebras for position-dependent mass Schr\"odinger
equations is highlighted by constructing spectrum generating algebras for a
class of d-dimensional radial harmonic oscillators with and a
specific mass choice depending on some positive parameter . Via some
minor changes, the one-dimensional oscillator on the line with the same kind of
mass is included in this class. The existence of a single unitary irreducible
representation belonging to the positive-discrete series type for and
of two of them for d=1 is proved. The transition to the constant-mass limit
is studied and deformed su(1,1) generators are constructed.
These operators are finally used to generate all the bound-state wavefunctions
by an algebraic procedure.Comment: 21 pages, no figure, 2 misprints corrected; published versio
Potential algebra approach to position dependent mass Schroedinger equation
It is shown that for a class of position dependent mass Schroedinger equation
the shape invariance condition is equivalent to a potential symmetry algebra.
Explicit realization of such algebras have been obtained for some shape
invariant potentials
Quasi-Hermitian supersymmetric extensions of a non-Hermitian oscillator Hamiltonian and of its generalizations
A harmonic oscillator Hamiltonian augmented by a non-Hermitian \pt-symmetric
part and its su(1,1) generalizations, for which a family of positive-definite
metric operators was recently constructed, are re-examined in a supersymmetric
context. Quasi-Hermitian supersymmetric extensions of such Hamiltonians are
proposed by enlarging su(1,1) to a
superalgebra. This allows the construction of new non-Hermitian Hamiltonians
related by similarity to Hermitian ones. Some examples of them are reviewed.Comment: 15 pages, no figure; published versio
A new family of shape invariantly deformed Darboux-P\"oschl-Teller potentials with continuous \ell
We present a new family of shape invariant potentials which could be called a
``continuous \ell version" of the potentials corresponding to the exceptional
(X_{\ell}) J1 Jacobi polynomials constructed recently by the present authors.
In a certain limit, it reduces to a continuous \ell family of shape invariant
potentials related to the exceptional (X_{\ell}) L1 Laguerre polynomials. The
latter was known as one example of the `conditionally exactly solvable
potentials' on a half line.Comment: 19 pages. Sec.5(Summary and Comments): one sentence added in the
first paragraph, several sentences modified in the last paragraph.
References: one reference ([25]) adde
The Localization of -Wave and Quantum Effective Potential of a Quasi-Free Particle with Position-Dependent Mass
The properties of the s-wave for a quasi-free particle with
position-dependent mass(PDM) have been discussed in details. Differed from the
system with constant mass in which the localization of the s-wave for the free
quantum particle around the origin only occurs in two dimensions, the
quasi-free particle with PDM can experience attractive forces in dimensions
except D=1 when its mass function satisfies some conditions. The effective mass
of a particle varying with its position can induce effective interaction which
may be attractive in some cases. The analytical expressions of the
eigenfunctions and the corresponding probability densities for the s-waves of
the two- and three-dimensional systems with a special PDM are given, and the
existences of localization around the origin for these systems are shown.Comment: 12pages, 8 figure
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