Using the pseudo-invariant operator method, we investigate the model of a particle with a time-dependent mass in a complex time-dependent symmetric potential well V (x, t) = if (t) |x|. The problem is exactly solvable and the analytic expressions of the Schrödinger wavefunctions are given in terms of the Airy function. Indeed, with an appropriate choice of the time-dependent metric operators and the unitary transformations, for each region, the two corresponding pseudo-Hermitian invariants transform into a well-known time-independent Hermitian invariant which is the Hamiltonian of a particle confined in a symmetric linear potential well. The eigenfunctions of the last invariant are the Airy functions. Then, the phases obtained are real for both regions and the general solution to the problem is deduced.Using the pseudo-invariant operator method, we investigate the model of a particle with a time-dependent mass in a complex time-dependent symmetric potential well V (x, t) = if (t) |x|. The problem is exactly solvable and the analytic expressions of the Schrödinger wavefunctions are given in terms of the Airy function. Indeed, with an appropriate choice of the time-dependent metric operators and the unitary transformations, for each region, the two corresponding pseudo-Hermitian invariants transform into a well-known time-independent Hermitian invariant which is the Hamiltonian of a particle confined in a symmetric linear potential well. The eigenfunctions of the last invariant are the Airy functions. Then, the phases obtained are real for both regions and the general solution to the problem is deduced.

arXiv:2206.04593Acta PolytechnicaMay 2, 20236 citations

Classical-quantum Interface of a Particle in a Time-dependent Linear Potential

L. Krache, Mustapha Maamache, Y. Saadi

International Journal of Theoretical PhysicsFeb 23, 20102 citations

Path integral solution for some time-dependent potential

S. N. Storchak

Physics Letters AJan 1, 19925 citations

On the pseudo-Hermitian invariant method for the time-dependent Non-Hermitian Hamiltonians

Boubakeur Khantoul, Abdelhafid Bounames, Mustapha Maamache

We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators $H(t)$ that generate a real phase in their time-evolution. This involves the use of invariant operators $I_{PH}(t)$ that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, $H(t)$ is not generally quasi-Hermitian and does not define an observable of the system but $I_{PH}(t)$ obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators $H(t)$ that generate a real phase in their time-evolution. This involves the use of invariant operators $I_{PH}(t)$ that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, $H(t)$ is not generally quasi-Hermitian and does not define an observable of the system but $I_{PH}(t)$ obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.

Oct 28, 20163 citations

Comment on: Quantum theory of a non-Hermitian time-dependent forced harmonic oscillator having 𝒫𝒯 symmetry

Naima Mana, Mustapha Maamache

Pedrosa et al.1 have recently used a [Formula: see text] symmetric linear invariant to study a unidimensional time-dependent [Formula: see text] symmetric harmonic oscillator with a complex time-dependent [Formula: see text] symmetric external force. We show in this comment that the normalization condition of the eigenfunctions of the invariant is not verified as claimed in Ref. 1. In order to obtain the normalization condition, we introduce a novel concept of the pseudoparity-time (pseudo-[Formula: see text]) symmetry.Pedrosa et al.1 have recently used a [Formula: see text] symmetric linear invariant to study a unidimensional time-dependent [Formula: see text] symmetric harmonic oscillator with a complex time-dependent [Formula: see text] symmetric external force. We show in this comment that the normalization condition of the eigenfunctions of the invariant is not verified as claimed in Ref. 1. In order to obtain the normalization condition, we introduce a novel concept of the pseudoparity-time (pseudo-[Formula: see text]) symmetry.

International Journal of Modern Physics AJan 10, 20203 citations

We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators

H(t)

that generate a real phase in their time-evolution. This involves the use of invariant operators

I_{PH}(t)

that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore,

H(t)

is not generally quasi-Hermitian and does not define an observable of the system but

I_{PH}(t)

obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.