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.
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].