17 research outputs found
Relativistic shape invariant potentials
Dirac equation for a charged spinor in electromagnetic field is written for
special cases of spherically symmetric potentials. This facilitates the
introduction of relativistic extensions of shape invariant potential classes.
We obtain the relativistic spectra and spinor wavefunctions for all potentials
in one of these classes. The nonrelativistic limit reproduces the usual
Rosen-Morse I & II, Eckart, Poschl-Teller, and Scarf potentials.Comment: Corrigendum: The last statement above equation (1) is now corrected
and replaced by two new statement
New approach to (quasi)-exactly solvable Schrodinger equations with a position-dependent effective mass
By using the point canonical transformation approach in a manner distinct
from previous ones, we generate some new exactly solvable or quasi-exactly
solvable potentials for the one-dimensional Schr\"odinger equation with a
position-dependent effective mass. In the latter case, SUSYQM techniques
provide us with some additional new potentials.Comment: 11 pages, no figur
Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass
Known shape-invariant potentials for the constant-mass Schrodinger equation
are taken as effective potentials in a position-dependent effective mass (PDEM)
one. The corresponding shape-invariance condition turns out to be deformed. Its
solvability imposes the form of both the deformed superpotential and the PDEM.
A lot of new exactly solvable potentials associated with a PDEM background are
generated in this way. A novel and important condition restricting the
existence of bound states whenever the PDEM vanishes at an end point of the
interval is identified. In some cases, the bound-state spectrum results from a
smooth deformation of that of the conventional shape-invariant potential used
in the construction. In others, one observes a generation or suppression of
bound states, depending on the mass-parameter values. The corresponding
wavefunctions are given in terms of some deformed classical orthogonal
polynomials.Comment: 26 pages, no figure, reduced secs. 4 and 5, final version to appear
in JP
Graded extension of SO(2,1) Lie algebra and the search for exact solutions of Dirac equation by point canonical transformations
SO(2,1) is the symmetry algebra for a class of three-parameter problems that
includes the oscillator, Coulomb and Morse potentials as well as other problems
at zero energy. All of the potentials in this class can be mapped into the
oscillator potential by point canonical transformations. We call this class the
"oscillator class". A nontrivial graded extension of SO(2,1) is defined and its
realization by two-dimensional matrices of differential operators acting in
spinor space is given. It turns out that this graded algebra is the
supersymmetry algebra for a class of relativistic potentials that includes the
Dirac-Oscillator, Dirac-Coulomb and Dirac-Morse potentials. This class is, in
fact, the relativistic extension of the oscillator class. A new point canonical
transformation, which is compatible with the relativistic problem, is
formulated. It maps all of these relativistic potentials into the
Dirac-Oscillator potential.Comment: Replaced with a more potrable PDF versio
Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem
We show that there exist some intimate connections between three
unconventional Schr\"odinger equations based on the use of deformed canonical
commutation relations, of a position-dependent effective mass or of a curved
space, respectively. This occurs whenever a specific relation between the
deforming function, the position-dependent mass and the (diagonal) metric
tensor holds true. We illustrate these three equivalent approaches by
considering a new Coulomb problem and solving it by means of supersymmetric
quantum mechanical and shape invariance techniques. We show that in contrast
with the conventional Coulomb problem, the new one gives rise to only a finite
number of bound states.Comment: 22 pages, no figure. Archive version is already official. Published
by JPA at http://stacks.iop.org/0305-4470/37/426
A class of solvable potentials
A systematic method of constructing (velocity-independent) potentials, for which the s-wave Schrüdinger equation can be solved in terms of known functions, is presented. Several such examples are constructed and the analytic structure of some of the corresponding scattering amplitudes worked out in detail. The use of Darboux's theorem allows a significant extension of the class of solvable potentials