# Sequences of closely spaced resonances and eigenvalues for bipartite complex potentials

## TL;DR

This paper analyzes the spectral properties of a Schroedinger operator with bipartite complex potentials separated by a large distance, revealing a sequence of resonances and eigenvalues near the real axis with potential applications in physics.

## Contribution

It introduces a novel analysis of the eigenvalues and resonances for bipartite complex potentials with large separation, highlighting their approximate equidistance and physical relevance.

## Key findings

- Existence of a sequence of approximately equidistant complex wavenumbers near the real axis.
- Wavenumbers correspond to either resonances or eigenvalues depending on their imaginary parts.
- Resonance sequence resembles transmission resonances in Fabry-Pérot interferometers.

## Abstract

We consider a Schroedinger operator on the axis with a bipartite potential consisting of two compactly supported complex-valued functions, whose supports are separated by a large distance. We show that this operator possesses a sequence of approximately equidistant complex-valued wavenumbers situated near the real axis. Depending on its imaginary part, each wavenumber corresponds to either a resonance or an eigenvalue. The obtained sequence of wavenumbers resembles transmission resonances in electromagnetic Fabry-P\'erot interferometers formed by parallel mirrors. Our result has potential applications in standard and non-hermitian quantum mechanics, physics of waveguides, photonics, and in other areas where the Schroedinger operator emerges as an effective Hamiltonian.