A non-Hermitian generalized oscillator model, generally known as the Swanson model, has been studied in the framework of R-deformed Heisenberg algebra. The non-Hermitian Hamiltonian is diagonalized by generalized Bogoliubov transformation. A set of deformed creation annihilation operators is introduced whose algebra shows that the transformed Hamiltonian has conformal symmetry. The spectrum is obtained using algebraic technique. The superconformal structure of the system is also worked out in detail. An anomaly related to the spectrum of the Hermitian counterpart of the non-Hermitian Hamiltonian with generalized ladder operators is shown to occur and is discussed in position dependent mass scenario.

http://arxiv.org/abs/1301.0716
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Using an appropriate change of variable, the Schroedinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi type \(X_m\) exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric and hyperbolic Scarf potentials whose bound state solutions are associated with the aforesaid exceptional orthogonal polynomials. These infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic Scarf potentials by the addition of some rational terms characterized by the presence of classical Jacobi polynomials. All the members of a particular family of these `rationally extended polynomial-dependent’ potentials have the same energy spectrum and possess translational shape invariant symmetry. The obtained non-Hermitian potentials are shown to be quasi-Hermitian in nature ensuring the reality of the associated energy spectra.

http://arxiv.org/abs/1210.0119
Mathematical Physics (math-ph); Quantum Physics (quant-ph)