A new effective mass Hamiltonian and associated Lamé equation: bound states

Abstract

Abstract.A new quantum model with rational functions for the potential and effective mass is proposed in a stretchable region outside which both are constant. Starting from a generalized effective mass kinetic energy operator the matching and boundary conditions for the envelope wave functions are derived. It is shown that in a mapping to an auxiliary constant-mass Schrödinger picture one obtains one-period "associated Lamé" well bounded by two δ-wells or δ-barriers depending on the values of the ordering parameter β. The results for bound states of this new solvable model are provided for a wide variation of the parameters.

Nonlinear dynamics of a position-dependent mass-driven Duffing-type oscillator

“…To summarize, we have demonstrated in this communication that given any value of f , if it is confronted with the PDM parameter ξ, remarkable phase transitions occur such as, for instance, from a limit cycle mode to a chaotic regime through a period doubling intermediate phase pointing to the sensitivity of the system (11) due to the interplay between f and ξ. The essential distinction between the constant mass and the variable mass case rests in the fact that the presence of the parameter ξ not only enhances the rapidity of such transitions but also initiates complicated nature of bifurcations, of course for a non-zero value of f .…”

SUSY method for the three-dimensional Schrödinger equation with effective mass

“…This problem is not quite trivial due to non-commutativity of the effective mass function m( x) with threevector of the momentum operator p = −i ∂ ≡ −i∂/∂ x. Different arguments exist for the choice of kinetic term (for example, see references in [70]), and we shall use the most popular (explicitly Hermitian) form [73]:…”

“…These models (also called as models with the effective mass) appear in many branches of physics very distinct from each other: quantum liquids [59], nuclear physics [61], quantum wires and dots [60], physics of semiconductors [62]- [65], and some others. The SUSY QM approach was used successfully to investigate the wide class of such models -see for example [66], [67], [68], [69], [70]. Although most of the papers on the effective mass were focused on one-dimensional systems, there were few two-dimensional [66], [67], [68], three-dimensional [69] and (2 + 1)-nonstationary [71] exclusions.…”

Position-dependent mass, finite-gap systems, and supersymmetry

“…See also refs. [32,33,34,35,36,37,38,39,40,41,42,43,44] where some other aspects of quantum systems with PDM were studied.…”