# Analytically solvable $2\times2$ $PT$-symmetry dynamics from su(1,1)-symmetry problems

## TL;DR

This paper presents a method to explicitly construct exact solutions for the time evolution of 2x2 PT-symmetric Hamiltonians, with applications to physical systems like coupled wave-guides.

## Contribution

It introduces a protocol for solving the dynamics of 2x2 PT-symmetric systems analytically, expanding the toolkit for studying non-Hermitian quantum systems.

## Key findings

- Explicit construction of time-evolution operators for PT-symmetric Hamiltonians
- Application examples demonstrating the method's effectiveness
- Discussion of physical relevance in gain-loss optical systems

## Abstract

A protocol for explicitly constructing the exact time-evolution operators generated by $2 \times 2$ time-dependent $PT$-symmetry Hamiltonians is reported. Its mathematical applicability is illustrated with the help of appropriate examples. The physical relevance of the proposed approach within gain-loss system scenarios, like two-coupled wave-guides, is discussed in detail.