Non-Hermitian von Roos Hamiltonian's $\eta$-weak-pseudo-Hermiticity and exact solvability

Omar Mustafa, S. Habib Mazharimousavi

Published Jul 25, 2007• 0 citations

Abstract

A complexied von Roos Hamiltonian is considered and a Hermitian rst-order intertwining di¤erential operator is used to obtain the related position dependent mass -weak-pseudo-Hermitian Hamiltonians. Two user -friendly reference-target maps are introduced to serve for exactsolvability of some non-Hermitian -weak-pseudo-Hermitian position dependent mass Hamiltonians. A non-Hermitian PT -symmetric Scarf II and a non-Hermitian periodic-type PT -symmetric Samsonov-Roy potentials are used as reference models in a user-friendly reference-target map and the corresponding isospectral Hamiltonians are obtained. It is observed that for each exactly-solvable reference Hamiltonian there is a corresponding set of exactly-solvable target Hamiltonians. PACS numbers: 03.65.Ge, 03.65.Fd, 03.65.Ca

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Non-Hermitian von Roos Hamiltonian's $\eta$ -weak-pseudo-Hermiticity and exact solvabilityNon-Hermitian von Roos Hamiltonian's $\eta$-weak-pseudo-Hermiticity and exact solvability

Omar Mustafa, S. Habib Mazharimousavi

Published Jul 25, 2007• 0 citations

OpenAlex

Abstract

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Book Publisher International (a part of SCIENCEDOMAIN International)Apr 20, 20220 citations

Non-Hermitian von Roos Hamiltonian's

\eta

-weak-pseudo-Hermiticity and exact solvability

\mathcal{PT}

Symmetric Hamiltonian Model and Exactly Solvable Potentials

Searching for non-Hermitian (parity-time)

\mathcal{PT}

-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a

\mathcal{PT}

symmetric non-Hermitian Hamiltonian model which is given as

\hat{\mathcal{H}}=ω(\hat{b}^\dagger\hat{b}+\frac{1}{2})+ α(\hat{b}^{2}-(\hat{b}^\dagger)^{2})

where

ω

and

α

are real constants,

\hat{b}

and

\hat{b^\dagger}

are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of

\mathcal{PT}

symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian

\mathcal{H}

is pseudo-Hermitian, we have obtained the Hermitian equivalent of

\mathcal{H}

which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential.

\mathcal{H}

is called pseudo-Hermitian, if there exists a Hermitian and invertible operator

η

satisfying

\mathcal{H^\dagger}=η\mathcal{H} η^{-1}

. For the Hermitian Hamiltonian

h

, one can write

h=ρ\mathcal{H} ρ^{-1}

where

ρ=\sqrtη

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.

A transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator

η

are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.

The non

P T-

symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field that couples to a harmonic oscillator via some pseudo-hermitian interaction is considered. We highlight all of the properties of both the original Mandal and Jaynes-Cummings Hamiltonians. We show that the Mandal Hamiltonian

H_{M}

is non hermitian and non invariant under the combined action of the parity operator P and the time-reversal operator T Even if the previous properties are not satisfied, it has been proved that the Mandal Hamiltonian

H_{M}

is pseudo-hermitian with respect to P and with respect to

\sigma_{3}

also [1,2]. Thus, we show that the original Jaynes-Cummings Hamiltonian is hermitian. We show the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator, similar to the direct approach to invariant vector spaces used in Refs [3,4].