# Spacing gain and absorption in a simple $\mathcal{PT}$-symmetric model: spectral singularities and ladders of eigenvalues and resonances

## TL;DR

This paper investigates a $ ext{PT}$-symmetric waveguide model, revealing how varying the gain-loss distance creates new spectral features and a ladder of resonances, with implications for laser-absorber modes and symmetry breaking.

## Contribution

It introduces a detailed analysis of spectral singularities and eigenvalue ladders in a $ ext{PT}$-symmetric waveguide with variable gain-loss separation, highlighting new phenomena not present in adjacent configurations.

## Key findings

- Increasing gain-loss distance creates new resonances and spectral singularities.
- A ladder of resonances emerges at large gain-loss separation.
- Resonances can be transformed into complex eigenvalues by adjusting amplitudes.

## Abstract

We consider a parity-time ($\mathcal{PT}$-) symmetric waveguide consisting of a localized gain and loss elements separated by a variable distance. The situation is modelled by a Schr\"odiner operator with localized complex $\mathcal{PT}$-symmetric potential. Properties of the latter Hamiltonian are considered subject to the change of the gain-to-loss distance. Resonances, spectral singularities and eigenvalues are analyzed in detail and discussed in the context of the associated laser-absorber modes and $\mathcal{PT}$-symmetry breaking phase transition. Increasing gain-to-loss distance creates new resonances and spectral singularities which do not exist in the waveguide with adjacent gain and loss. In the limit of large gain-to-loss distance, the waveguide features a ladder of resonances which can be transformed to a ladder of complex eigenvalues by means of the change of the gain-and-loss amplitude.