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