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New exact and approximation methods for time-dependent non-Hermitian quantum systems

Tenney, R. (2022). New exact and approximation methods for time-dependent non-Hermitian quantum systems. (Unpublished Doctoral thesis, City, University of London)

Abstract

The focus of this thesis is new methods, approximate and exact, in the areas of time-dependent Hermitian and non-Hermitian quantum mechanics.

By utilising the Lewis-Riesenfeld method of invariants we first present an approach which makes use of time-independent approximations such as standard time-independent perturbation theory and WKB theory to provide solutions to the time-dependent Schrodinger equation. The validity of the method is illustrated in
its application to the study of two exactly solvable Hermitian systems, the timedependent harmonic oscillator with Stark term and the Goldman-Krivchenko potential with a time-dependent perturbation.

Our focus then shifts to non-Hermitian systems where we present the first exact solution to the time-dependent Dyson equation for the time-dependent anharmonic quartic oscillator demonstrating that it is spectrally equivalent to a time-dependent double well potential.

To aid in the construction of time-dependent Dyson maps and metrics we employ point transformations connecting time-dependent non-Hermitian systems with stationary Hermitian ones to compute exact invariants. Here we study the time-dependent Swanson model and the time-dependent harmonic oscillator with complex linear potential. The approach is further applied to the time-dependent anharmonic quartic oscillator for which we present a second solution to the time-dependent Dyson equation.

A perturbative scheme for finding the time-dependent Dyson map and metric is then proposed and applied to determine exact solutions for a pair of weakly coupled two dimensional time-dependent harmonic oscillators with non-Hermitian coupling in space and momenta and the strongly coupled time-dependent anharmonic oscillator. We also consider two-dimensional time-dependent harmonic oscillators where the non-Hermitian coupling is just in momenta.

Finally we explore a procedure which allows for systematic production of an infinite series of time-dependent Dyson maps governed by the symmetries of the Lewis-Riesenfeld invariants for time-dependent non-Hermitian Hamiltonians and their equivalent Hermitian Hamiltonians. We find an infinite number of solutions for the aforementioned harmonic oscillators with non-Hermitian space and momenta coupling as well as the time-dependent anharmonic oscillator.

Publication Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
Departments: Doctoral Theses
Doctoral Theses > School of Science & Technology Doctoral Theses
School of Science & Technology > Mathematics
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