19 research outputs found

### Potentials with Two Shifted Sets of Equally Spaced Eigenvalues and Their Calogero Spectrum

Motivated by the concept of shape invariance in supersymmetric quantum
mechanics, we obtain potentials whose spectrum consists of two shifted sets of
equally spaced energy levels. These potentials are similar to the
Calogero-Sutherland model except the singular term $\alpha x^{-2}$ always falls
in the transition region $-1/4 < \alpha < 3/4$ and there is a delta-function
singularity at x=0.Comment: Latex, 12 pages, Figures available from Authors, To appear in Physics
Letters A. Please send requests for figures to [email protected] or
[email protected]

### Methods for Generating Quasi-Exactly Solvable Potentials

We describe three different methods for generating quasi-exactly solvable
potentials, for which a finite number of eigenstates are analytically known.
The three methods are respectively based on (i) a polynomial ansatz for wave
functions; (ii) point canonical transformations; (iii) supersymmetric quantum
mechanics. The methods are rather general and give considerably richer results
than those available in the current literature.Comment: 12 pages, LaTe

### Non-Central Potentials and Spherical Harmonics Using Supersymmetry and Shape Invariance

It is shown that the operator methods of supersymmetric quantum mechanics and
the concept of shape invariance can profitably be used to derive properties of
spherical harmonics in a simple way. The same operator techniques can also be
applied to several problems with non-central vector and scalar potentials. As
examples, we analyze the bound state spectra of an electron in a Coulomb plus
an Aharonov-Bohm field and/or in the magnetic field of a Dirac monopole.Comment: Latex, 12 pages. To appear in American Journal of Physic

### Exact solution of a class of three-body scattering problems in one dimension

We present an exact solution of the three-body scattering problem for a one-parameter family of one-dimensional potentials containing the Calogero and Wolfes potentials as special limiting cases. The result is an interesting nontrivial relationship between the final momenta p'i and the initial momenta pi of the three particles. We also discuss another one-parameter family of potentials for all of which p'i = -pi (i = 1,2,3)

### Local Identities Involving Jacobi Elliptic Functions

We derive a number of local identities of arbitrary rank involving Jacobi
elliptic functions and use them to obtain several new results. First, we
present an alternative, simpler derivation of the cyclic identities discovered
by us recently, along with an extension to several new cyclic identities of
arbitrary rank. Second, we obtain a generalization to cyclic identities in
which successive terms have a multiplicative phase factor exp(2i\pi/s), where s
is any integer. Third, we systematize the local identities by deriving four
local ``master identities'' analogous to the master identities for the cyclic
sums discussed by us previously. Fourth, we point out that many of the local
identities can be thought of as exact discretizations of standard nonlinear
differential equations satisfied by the Jacobian elliptic functions. Finally,
we obtain explicit answers for a number of definite integrals and simpler forms
for several indefinite integrals involving Jacobi elliptic functions.Comment: 47 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

### Linear Superposition in Nonlinear Equations

Even though the KdV and modified KdV equations are nonlinear, we show that
suitable linear combinations of known periodic solutions involving Jacobi
elliptic functions yield a large class of additional solutions. This procedure
works by virtue of some remarkable new identities satisfied by the elliptic
functions.Comment: 7 pages, 1 figur

### Periodic Solutions of Nonlinear Equations Obtained by Linear Superposition

We show that a type of linear superposition principle works for several
nonlinear differential equations. Using this approach, we find periodic
solutions of the Kadomtsev-Petviashvili (KP) equation, the nonlinear
Schrodinger (NLS) equation, the $\lambda \phi^4$ model, the sine-Gordon
equation and the Boussinesq equation by making appropriate linear
superpositions of known periodic solutions. This unusual procedure for
generating solutions is successful as a consequence of some powerful, recently
discovered, cyclic identities satisfied by the Jacobi elliptic functions.Comment: 19 pages, 4 figure