C V Sukumar

SUPERSYMMETRIC QUANTUM MECHANICS AND THE INVERSE SCATTERING METHOD

The procedures for finding a new potential (1) by eliminating the ground state of a given potential, (2) by adding a bound state below the ground state of a given potential and (3) by generating the phase equivalent family of a given potential using the supersymmetric pairing of the spectra of the operators A+A- and A-A+ are compared with the application of the Gelfand-Levitan procedure (1955) for the corresponding cases. It is shown how the equivalence of the two procedures may be established. A distinction is made between the modifications of the Jost functions associated with four different types of transformations generated by the concept of a supersymmetric partner to a given Schrodinger equation. It is shown that the Bargmann class of potentials may be generated using suitable combinations of the four types of transformations.

Some soluble models for periodic decay and revival

The authors present three analytically tractable models for a two-level atom interacting with a coherent or thermal radiation field and study the behaviour of the atomic excitation energy for various initial conditions.

Supersymmetry and the Dirac equation for a central Coulomb field

It is shown that the methods of supersymmetric quantum mechanics can be used to obtain the complete energy spectrum and eigenfunctions of the Dirac equation for an attractive Coulomb potential.

Solution of the Heisenberg equations for an atom interacting with radiation

It is shown that the nonlinear equation of motion for the energy operator of a few-level atom interacting with a single mode radiation field can be solved explicitly.

Supersymmetry, potentials with bound states at arbitrary energies and multi-soliton configurations

The connection between the algebra of supersymmetry and the inverse scattering method is used to construct one-dimensional potentials with any specified number of non-degenerate bound states at arbitrary energies. The reflection coefficient of the potential so constructed is related to the reflection coefficient of a reference potential which supports no bound states. It is shown that, by choosing the reference potential to be V=0, it is possible to construct reflectionless potentials with bound states at arbitrary energies. The relationship of this construction based on supersymmetry to other known constructions of reflectionless potentials is established. It is shown that the symmetric reflectionless potential may be expressed as a linear combination of the squares of bound state eigenfunctions with coefficients related to the wavenumbers associated with the bound states.

Supersymmetry and potentials with bound states at arbitrary energies. II

For pt.I see ibid vol.19, p.2297, 1986. It has been shown previously that a potential V0( chi ) in one dimension which supports no bound states may be used as a reference potential from which, by successive applications of the concept of a supersymmetric partner to a given Hamiltonian, it is possible to find a potential Vn( chi ) which supports any specified number n of bound states at any chosen energies Ej, j=1,. . .,n. The reflection coefficient of Vn is related to the reflection coefficient of V0. Various alternative representations of the potentials constructed by this procedure are presented. An illustrative example in which Vn is constructed by using a sech2 chi barrier as the reference potential is discussed.

Supersymmetric quantum mechanics and its applications

The Hamiltonian in Supersymmetric Quantum Mechanics is defined in terms of charges that obey the same algebra as that of the generators of supersymmetry in field theory. The consequences of this symmetry for the spectra of the component parts that constitute the supersymmetric system are explored. The implications of supersymmetry for the solutions of the Schrodinger equation, the Dirac equation, the inverse scattering theory and the multi‐soliton solutions of the KdV equation are examined. Applications to scattering problems in Nuclear Physics with specific reference to singular potentials which arise from considerations of supersymmetry will be discussed.

Green's functions, sum rules and matrix elements for SUSY partners

It is now well known that the solutions in the Schrodinger equations for two potentials which are supersymmetric partners are linked by intertwining relations involving first-order differential operators. In this paper, we explore the consequence of this linkage for the Greens functions corresponding to SUSY partners and the relation between the sums over the inverses of the eigenvalues for the two potentials. We also establish some relations between the matrix elements of certain operators evaluated between the eigenstates of two partner potentials. We show that there can be circumstances where some matrix elements vanish as a consequence of the presence of supersymmetry.

SUSY transformation of the Green function and a trace formula

An integral relation is established between the Green functions corresponding to two Hamiltonians which are supersymmetric (SUSY) partners and in general may possess both discrete and continuous spectra. It is shown that when the continuous spectrum is present, the trace of the difference of the Green functions for SUSY partners is a finite quantity which may or may not be equal to zero despite the divergence of the traces of each Green function. Our findings are illustrated by using the free particle example considered both on the whole real line and on a half line.

Potentials generated by SU(1,1)

A systematic procedure for deriving a class of potentials with the underlying symmetry group SU(1,1), starting the commutation relations for the generators of SU(1,1), is presented.