Sequences of closely spaced resonances and eigenvalues for bipartite complex potentials

We consider a Schrödinger 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–Pérot 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 Schrödinger operator emerges as an effective Hamiltonian.