Asish Ganguly

A study of the bound states for square potential wells with position-dependent mass

A potential well with position-dependent mass is studied for bound states. Applying appropriate matching conditions, a transcendental equation is derived for the energy eigenvalues. Numerical results are presented graphically and the variation of the energy of the bound states are calculated as a function of the well-width and mass.

Shape-invariant quantum Hamiltonian with position-dependent effective mass through second-order supersymmetry

A second-order supersymmetric approach is taken to the system describing motion of a quantum particle in a potential endowed with position-dependent effective mass. It is shown that the intertwining relations between second-order partner Hamiltonians may be exploited to obtain a simple shape-invariant condition. Indeed, a novel relation between potential and mass functions is derived, which leads to a class of exactly solvable models. As an illustration of our procedure, two examples are given for which one obtains whole spectra algebraically. Both shape-invariant potentials exhibit harmonic-oscillator-like or singular-oscillator-like spectra depending on the values of the shape-invariant parameter.

Higher derivative supersymmetry, a modified Crum-Darboux transformation and coherent state

Inspired by the possibility of factorizing the second derivative interwining operators using a modified form of the well-known Crum–Darboux transformation, we present a scheme for generating a new pair of isospectral potentials. We also consider the interesting problem of constructing coherent states for such factorizable operators.

Exactly solvable associated Lamé potentials and supersymmetric transformations

A systematic procedure to derive exact solutions of the associated Lame equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are used to generate new exactly solvable potentials; some of them exhibit an interesting property of periodicity defects.

New exact solutions of a generalized shallow water wave equation

In this work, an extended elliptic function method is proposed and applied to the generalized shallow water wave equation. We systematically investigate on how to classify new exact travelling wave solutions expressible in terms of quasi-periodic elliptic integral functions and doubly periodic Jacobian elliptic functions. The derived new solutions include rational, periodic, singular and solitary wave solutions. An interesting comparison with the canonical procedure is provided. In some cases the obtained elliptic solution has singularity at a certain region in the whole space. For such solutions we have computed the effective region where the obtained solution is free from such a singularity.

A unified treatment of exactly solvable and quasi-exactly solvable quantum potentials

By exploiting the hidden algebraic structure of the Schrodinger Hamiltonian, namely the sl(2), we propose a unified approach of generating both exactly solvable and quasi-exactly solvable potentials. We obtain, in this way, two new classes of quasi-exactly solvable systems one of which is of periodic type while the other hyperbolic.

ASSOCIATED LAMÉ EQUATION, PERIODIC POTENTIALS AND sl(2, ℝ)

We propose a new approach based on the algebraization of the associated Lame equation within sl(2, ℝ) to derive the corresponding periodic potentials. The band edge eigenfunctions and energy spectra are explicitly obtained for integers m, l. We also obtain the explicit expressions of the solutions for half-integer m and integer or half-integer l.

Associated Lamé and various other new classes of elliptic potentials from sl(2,R) and related orthogonal polynomials

Using representations of sl(2,R) generators which yield associated Lame Hamiltonians we obtain new classes of elliptic potentials. We explicitly calculate eigenstates and spectra for these potentials and construct the associated orthogonal polynomials. We show that in the proper limit these potentials reduce to well-known exactly solvable potentials.

Supersymmetry across nanoscale heterojunction

We argue that supersymmetric transformation could be applied across the heterojunction formed by joining of two mixed semiconductors. A general framework is described by specifying the structure of ladder operators at the junction for making quantitative estimation of physical quantities. For a particular heterojunction device, we show that an exponential grading inside a nanoscale doped layer is amenable to exact analytical treatment for a class of potentials distorted by the junctions through the solutions of transformed Morse-type potentials.

Supersymmetric partners for the associated Lamé potentials

The general solution of the stationary Schrödinger equation for the associated Lamé potentials with an arbitrary real energy is found. The supersymmetric partners are generated by employing seed solutions for factorization energies inside the gaps.