Non-Hermitian Hamiltonians appearing as operator realizations of -symmetric (-symmetric in physical literature) zero-range singular perturbations of one-dimensional Schrodinger operators are studied. In particular, Hamiltonians with a real spectrum are described in terms of parameters of singular perturbations and, moreover, it is shown that only part of them are similar to Hermitian ones. In this case, they can be used as exactly solvable models of -symmetric quantum mechanics.
Non-Hermitian Hamiltonians appearing as operator realizations of -symmetric (-symmetric in physical literature) zero-range singular perturbations of one-dimensional Schrodinger operators are studied. In particular, Hamiltonians with a real spectrum are described in terms of parameters of singular perturbations and, moreover, it is shown that only part of them are similar to Hermitian ones. In this case, they can be used as exactly solvable models of -symmetric quantum mechanics.
Searching for non-Hermitian, -symmetric Hamiltonians [1] with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a symmetric non-Hermitian Hamiltonian model which is given as where ω and α are real constants, and are first order differential operators. Because the Hamiltonian is pseudo-Hermitian, we have obtained the Hermitian equivalent of which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened and hyperbolic Rosen-Morse II potentials. Using convenient sinilarity transformations, we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics [2].
PT symmetric Hamiltonian model and Dirac equation in 1+1 dimensions
In this paper, we have introduced a symmetric non-Hermitian Hamiltonian model which is given as where ω and α are real constants, and are first-order differential operators. The Hermitian form of the Hamiltonian is obtained by suitable mappings and it is interrelated to the time-independent one-dimensional Dirac equation in the presence of position-dependent mass. Then, Dirac equation is reduced to a Schrödinger-like equation and two new complex non- symmetric vector potentials are generated. We have obtained a real spectrum for these new complex vector potentials using the shape invariance method. We have searched the real energy values using numerical methods for the specific values of the parameters.
PT Symmetric Hamiltonian Model and Exactly Solvable Potentials
Searching for non-Hermitian (parity-time)PT-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a PT symmetric non-Hermitian Hamiltonian model which is given as H^=ω(b^†b^+21)+α(b^2−(b^†)2) where ω and α are real constants, b^ and b†^ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of PT symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian H is pseudo-Hermitian, we have obtained the Hermitian equivalent of H which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. H is called pseudo-Hermitian, if there exists a Hermitian and invertible operator η satisfying H†=ηHη−1. For the Hermitian Hamiltonian h, one can write h=ρHρ−1 where ρ=η is unitary. Using this ρ we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function ρ is obtained for each potential case.
The spectrum of a one-dimensional Hamiltonian with potential V(x)=ix2 for negative x and V(x)=−ix2 for positive x is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.