S. Sree Ranjani

BOUND STATE WAVE FUNCTIONS THROUGH THE QUANTUM HAMILTON–JACOBI FORMALISM

The bound state wave functions for a wide class of exactly solvable potentials are found by utilizing the quantum Hamilton–Jacobi formalism of Leacock and Padgett. It is shown that, exploiting the singularity structure of the quantum momentum function, until now used only for obtaining the bound state energies, one can straightforwardly find both the eigenvalues and the corresponding eigenfunctions. After demonstrating the working of this approach through a few solvable examples, we consider Hamiltonians, which exhibit broken and unbroken phases of supersymmetry. The natural emergence of the eigenspectra and the wave functions, in both unbroken and the algebraically nontrivial broken phase, demonstrates the utility of this formalism.

A study of quasi-exactly solvable models within the quantum Hamilton?Jacobi formalism

A few quasi-exactly solvable models are studied within the quantum Hamilton–Jacobi formalism. By assuming a simple singularity structure of the quantum momentum function, we show that the exact quantization condition leads to the condition for quasi-exact solvability.

Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition

We study the quantum Hamilton–Jacobi (QHJ) equation of the recently obtained exactly solvable models, related to the newly discovered exceptional polynomials, and show that the QHJ formalism reproduces the exact eigenvalues and the eigenfunctions. The fact that the eigenfunctions have zeros and poles in complex locations leads to an unconventional singularity structure of the quantum momentum function p(x), the logarithmic derivative of the wavefunction, which forms the crux of the QHJ approach to quantization. A comparison of the singularity structure for these systems with the known exactly solvable and quasi-exactly solvable models reveals interesting differences. We find that the singularity structure of the momentum function for these new potentials lies between the above two distinct models, sharing similarities with both of them. This prompted us to examine the exactness of the supersymmetric Wentzel–Kramers–Brillouin (SWKB) quantization condition. The interesting singularity structure of p(x) and of the superpotential for these models has important consequences for the SWKB rule and in our proof of its exactness for these quantal systems.

BAND EDGE EIGENFUNCTIONS AND EIGENVALUES FOR PERIODIC POTENTIALS THROUGH THE QUANTUM HAMILTON–JACOBI FORMALISM

We demonstrate the procedure of finding the band edge eigenfunctions and eigenvalues of periodic potentials, through the quantum Hamilton–Jacobi formalism. The potentials studied here are the Lame and associated Lame, which belong to the class of elliptic potentials. The formalism requires an assumption about the singularity structure of the quantum momentum function p, obeying a Riccati type equation in the complex x-plane. Essential use is made of suitable conformal transformations, which lead to the eigenvalues and the eigenfunctions corresponding to the band edges, in a straightforward manner. Our study reveals interesting features about the singularity structure of p, underlying the band edge states.

QUANTUM HAMILTON–JACOBI ANALYSIS OF PT SYMMETRIC HAMILTONIANS

We apply the quantum Hamilton–Jacobi formalism, naturally defined in the complex domain, to complex Hamiltonians, characterized by discrete parity and time reversal (PT) symmetries and obtain their eigenvalues and eigenfunctions. Examples of both quasi-exactly and exactly solvable potentials are analyzed and the subtle differences, in the singularity structures of their quantum momentum functions, are pointed out. The role of the PT symmetry in the complex domain is also illustrated.

Periodic Quasi-Exactly Solvable Models

Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lamé potential, are shown to emerge naturally in the quantum Hamilton–Jacobi (QHJ) approach. We study the singularity structure of the quantum momentum function, which yields the band-edge eigenvalues and eigenfunctions and compare it with the solvable and quasi-exactly solvable non-periodic potentials, as well as the periodic ones.

Bound states and band structure : A unified treatment through the quantum Hamilton-Jacobi approach

Abstract We analyze the Scarf potential, which exhibits both discrete energy bound states and energy bands, through the quantum Hamilton–Jacobi approach. The singularity structure and the boundary conditions in the above approach, naturally isolate the bound and periodic states, once the problem is mapped to the zero energy sector of another quasi-exactly solvable quantum problem. The energy eigenvalues are obtained without having to solve for the corresponding eigenfunctions explicitly. We also demonstrate how to find the eigenfunctions through this method.

Soliton response to transient trap variations

The response of bright and dark solitons to rapid variations in an expulsive longitudinal trap is investigated. We concentrate on the effect of transient changes in the trap frequency in the form of temporal delta kicks and the hyperbolic cotangent functions. Exact expressions are obtained for the soliton profiles. This is accomplished using the fact that a suitable linear Schr?dinger stationary state solution in time can be effectively combined with the solutions of the nonlinear Schr?dinger equation, for obtaining solutions of the Gross?Pitaevskii equation with a time-dependent scattering length in a harmonic trap. Interestingly, there is rapid pulse amplification in certain scenarios.

Construction of localized atomic wave packets

It is shown that highly localized solitons can be created in lower dimensional Bose–Einstein condensates (BECs), trapped in a regular harmonic trap, by temporally varying the trap frequency. A BEC confined in such a trap can be effectively used to construct a pulsed atomic laser emitting coherent atomic wave packets. In addition to having a complete control over the spatio-temporal dynamics of the solitons, we can separate the equation governing the Kohn mode (centre of mass motion). We investigate the effect of the temporal modulation of the trap frequency on the spatio-temporal dynamics of the bright solitons and also on the Kohn mode. The dynamics of the solitons and the variations in the Kohn mode with time are compared with those in a BEC confined in a trap with unmodulated trap frequency.

An explicit realization of fractional statistics in one dimension

Abstract An explicit realization of anyons is provided, using the three-body Calogero model. The fact that in the coupling domain, - 1 / 4 g 0 , the angular spectrum can have a band structure, leads to the manifestation of the desired phase in the wave function, under the exchange of particles. Concurrently, the momentum corresponding to the angular variable is quantized, exactly akin to the relative angular momentum quantization in two dimensional anyonic system.