Spectral properties of confining superexponential potentials

We explore the spectral properties and behaviour of confining superexponential potentials. Several prototypes of these highly nonlinear potentials are analysed in terms of the eigenvalues and eigenstates of the underlying stationary Schrodinger equation up to several hundreds of excited states. A generalization of the superexponential self-interacting oscillator shows a scaling behaviour of the spacing of the eigenvalues which turns into an alternating behaviour for the power law modified oscillator. Superexponential potentials with an oscillating power show a very rich spectral structure with varying amplitudes and wave vectors. In the parity symmetric case doublets of near degenerate energy eigenvalues emerge in the spectrum. The corresponding eigenstates are strongly localized in the outer wells of the potential and occur as even-odd pairs which are interspersed into the spectrum of delocalized states. We provide an outlook on future perspectives including the possibility to use these features for applications in e.g. cold atom physics.

Automated summary

The spectral properties and behavior of confining superexponential potentials were explored, analyzing various prototypes in terms of eigenvalues and eigenstates. The self-interacting oscillator showed scaling behavior while the power law modified oscillator had alternating behavior. Superexponential potentials with an oscillating power exhibited a rich spectral structure with localized eigenstates and alternating even-odd pairs in the spectrum. Perspectives for future applications, including cold atom physics, were discussed.