Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians

TL;DR

This paper develops explicit solutions for a broad class of time-dependent non-Hermitian quantum systems with PT-symmetry, demonstrating quasi-exact solvability through the construction of Lewis-Riesenfeld invariants in a metric framework.

Contribution

It introduces a method to solve time-dependent non-Hermitian Hamiltonians using Lie algebra generators and metric-based invariants, extending solvability to explicitly time-dependent models.

Findings

Explicit solutions for PT-symmetric, time-dependent Hamiltonians.

Identification of a quasi-exactly solvable Hermitian model.

Construction of Lewis-Riesenfeld invariants in the metric picture.

Abstract

For a large class of time-dependent non-Hermitain Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis-Riesenfeld invariants making useof the metric picture, which is an equivalent alternative to the Schr\"{o}dinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian.