Exact solutions for time-dependent complex symmetric potential well

Boubakeur Khantoul; Abdelhafid Bounames

doi:10.14311/AP.2023.63.0132

Authors

Boubakeur Khantoul University of Jijel, Department of Physics, Laboratory of Theoretical Physics, BP 98 Ouled Aissa, 18000 Jijel, Algeria; University of Constantine 3 – Salah Boubnider University, Department of Process Engineering, BP B72 Ali Mendjeli, 25000 Constantine, Algeria https://orcid.org/0000-0001-9012-4864
Abdelhafid Bounames University of Jijel, Department of Physics, Laboratory of Theoretical Physics, BP 98 Ouled Aissa, 18000 Jijel, Algeria https://orcid.org/0000-0002-5998-2710

DOI:

https://doi.org/10.14311/AP.2023.63.0132

Keywords:

non-Hermitian Hamiltonian, time-dependent Hamiltonian, pseudo-invariant method, PT-symmetry, pseudo-Hermiticity

Abstract

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.

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