17 research outputs found
Generation of a Novel Exactly Solvable Potential
We report a new shape invariant (SI) isospectral extension of the Morse
potential. Previous investigations have shown that the list of "conventional"
SI superpotentials that do not depend explicitly on Planck's constant
is complete. Additionally, a set of "extended" superpotentials has been
identified, each containing a conventional superpotential as a kernel and
additional -dependent terms. We use the partial differential equations
satisfied by all SI superpotentials to find a SI extension of Morse with novel
properties. It has the same eigenenergies as Morse but different asymptotic
limits, and does not conform to the standard generating structure for
isospectral deformations.Comment: 9 pages, 3 figure
Generation of a Complete Set of Supersymmetric Shape Invariant Potentials from an Euler Equation
In supersymmetric quantum mechanics, shape invariance is a sufficient
condition for solvability. We show that all conventional additive shape
invariant superpotentials that are independent of obey two partial
differential equations. One of these is equivalent to the one-dimensional Euler
equation expressing momentum conservation for inviscid fluid flow, and it is
closed by the other. We solve these equations, generate the set of all
conventional shape invariant superpotentials, and show that there are no others
in this category. We then develop an algorithm for generating all additive
shape invariant superpotentials including those that depend on
explicitly.Comment: 4 page
Broken Supersymmetric Shape Invariant Systems and Their Potential Algebras
Although eigenspectra of one dimensional shape invariant potentials with
unbroken supersymmetry are easily obtained, this procedure is not applicable
when the parameters in these potentials correspond to broken supersymmetry,
since there is no zero energy eigenstate. We describe a novel two-step shape
invariance approach as well as a group theoretic potential algebra approach for
solving such broken supersymmetry problems.Comment: Latex file, 10 page
Method for Generating Additive Shape Invariant Potentials from an Euler Equation
In the supersymmetric quantum mechanics formalism, the shape invariance
condition provides a sufficient constraint to make a quantum mechanical problem
solvable; i.e., we can determine its eigenvalues and eigenfunctions
algebraically. Since shape invariance relates superpotentials and their
derivatives at two different values of the parameter , it is a non-local
condition in the coordinate-parameter space. We transform the shape
invariance condition for additive shape invariant superpotentials into two
local partial differential equations. One of these equations is equivalent to
the one-dimensional Euler equation expressing momentum conservation for
inviscid fluid flow. The second equation provides the constraint that helps us
determine unique solutions. We solve these equations to generate the set of all
known -independent shape invariant superpotentials and show that there
are no others. We then develop an algorithm for generating additive shape
invariant superpotentials including those that depend on explicitly,
and derive a new -dependent superpotential by expanding a Scarf
superpotential.Comment: 1 figure, 4 tables, 18 page
The Jewishness of Jesus and Christian-Jewish Dialogue: A Colloquium in Honor of Joep van Beeck, S.J.
No abstract is available