Solvable relativistic quantum dots with vibrational spectra

Abstract

For Klein-Gordon equation a consistent physical interpretation of wave functions is reviewed as based on a proper modification of the scalar product in Hilbert space. Bound states are then studied in a deep-square-well model where the spectrum is roughly equidistant and where a fine-tuning of the levels is mediated by $\mathcal{P}\mathcal{T}$-symmetric interactions (composed of imaginary delta functions) which mimic creation/annihilation processes.