Anjana Sinha

PT symmetry of a conditionally exactly solvable potential

Abstract A conditionally exactly solvable potential, the supersymmetric partner of the harmonic oscillator is investigated in the PT -symmetric setting. It is shown that a number of properties characterizing shape-invariant and Natanzon-class potentials generated by an imaginary coordinate shift x−iϵ also hold for this potential outside the Natanzon class.

An exactly solvable PT symmetric potential from the Natanzon class

The symmetric version of the generalized Ginocchio potential, a member of the general exactly solvable Natanzon potential class, is analysed and its properties are compared with those of symmetric potentials from the more restricted shape-invariant class. It is found that the symmetric generalized Ginocchio potential has a number of properties in common with the latter potentials: it can be generated by an imaginary coordinate shift x → x + ie, its states are characterized by the quasi-parity quantum number, the spontaneous breakdown of symmetry occurs at the same time for all the energy levels and it has two supersymmetric partners which cease to be symmetric when the symmetry of the original potential is spontaneously broken.

Generation of Exactly Solvable Non-Hermitian Potentials with Real Energies

Series of exactly solvable non-trivial complex potentials (possessing real spectra) are generated by applying the Darboux transformation to the excited eigenstates of a non-Hermitian potential V(x). This method yields an infinite number of non-trivial partner potentials, defined over the whole real line, whose spectra are nearly exactly identical to the original potential.

Isospectral partners of a complex PT-invariant potential

Abstract We construct isospectral partner potentials of a complex PT -invariant potential, viz., V(x)=−V1sech2x−iV2sechxtanhx using Darbouxs method. One set of isospectral potentials are obtained which can be termed ‘satellite potentials’, in the sense that they are of the same form as the original potential. In a particular case, the supersymmetric partner potential has the same spectrum, including the zero energy ground state, a fact which cannot occur in conventional supersymmetric quantum mechanics with real potential. An explicit example of a non-trivial set of isospectral potential is also obtained.

Wentzel–Kramers–Brillouin quantization rules for two-dimensional quantum dots

In this paper, we apply the semiclassical Wentzel–Kramers–Brillouin (WKB) approximation to an electron in a central force potential, confined in a two-dimensional disc, and we obtain the quantization rules for such a system. As explicit examples, we consider the two most widely studied potentials, viz., the parabolic potential (the harmonic oscillator) and the Hydrogenic impurity state (Coulomb potential) in two dimensions. Both the systems are confined within an impenetrable circular wall of radius r0: In particular, we determine the energies as well as eigenfunctions by this approach. On comparing these energies with those from exact numerical values, the agreement is found to be quite good. Moreover, the WKB approach is found to give a good estimate of the wave functions as well. These results suggest that the WKB approximation works well even for such rigid wall spatial confinement and the present approach can be applied to other confined systems such as those which are encountered in mesoscopic physics. r 2002 Elsevier Science B.V. All rights reserved.

New exactly solvable isospectral partners for symmetric potentials

We examine, in detail, the possibility of applying the Darboux transformation to non-Hermitian Hamiltonians. In particular we propose a simple method of constructing exactly solvable symmetric potentials by applying Darboux transformation to higher states of an exactly solvable symmetric potential. It is shown that the resulting Hamiltonian and the original one are pseudo supersymmetric partners. We also discuss application of the Darboux transformation to Hamiltonians with spontaneously broken symmetry.

Study of classical mechanical systems with complex potentials

Abstract We apply the factorization technique developed by Kuru and Negro [Ann. Phys. 323 (2008) 413] to study complex classical systems. As an illustration we apply the technique to study the classical analogue of the exactly solvable PT symmetric Scarf II model, which exhibits the interesting phenomenon of spontaneous breakdown of PT symmetry at some critical point. As the parameters are tuned such that energy switches from real to complex conjugate pairs, the corresponding classical trajectories display a distinct characteristic feature — the closed orbits become open ones.

Scattering states of a particle, with position-dependent mass, in a double heterojunction

In this work we obtain the exact analytical scattering solutions of a particle (electron or hole) in a semiconductor double heterojunction —potential well/barrier— where the effective mass of the particle varies with position inside the heterojunctions. It is observed that the spatial dependence of mass within the well/barrier introduces a nonlinear component in the plane-wave solutions of the continuum states. Additionally, the transmission coefficient is found to increase with increasing energy, finally approaching unity, whereas the reflection coefficient follows the reverse trend and goes to zero.

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.

Classical trajectories of the continuum states of the \chem{\cal{PT}} symmetric Scarf II potential

We apply the factorization technique developed by Kuru et. al. [Ann. Phys. {\bf 323} (2008) 413] to obtain the exact analytical classical trajectories and momenta of the continuum states of the non Hermitian but