Exact solutions for time-dependent complex symmetric potential well

TL;DR

This paper provides exact analytical solutions for a quantum particle with time-dependent mass in a complex symmetric potential well, using pseudo-invariant operators and Airy functions, revealing real phases and explicit wavefunctions.

Contribution

It introduces an exact solution method for a time-dependent complex potential well using pseudo-invariant operators and transforms to Hermitian invariants, which is novel.

Findings

Analytic wavefunctions expressed in Airy functions

Real phases obtained for the solutions

Exact solvability of the time-dependent complex potential

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 $V\left( x,t\right) =if\left(t\right) \left\vert x\right\vert$. The problem is exactly solvable and the analytic expressions of the Schr\"{o}dinger 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.