The continuous spectrum of bound states in expulsive potentials

TL;DR

This paper reveals that steep expulsive potentials can support normalizable eigenstates forming a continuous spectrum, challenging the intuition that such potentials lead to delocalization.

Contribution

It demonstrates the existence of normalizable bound states in steep expulsive potentials in 1D and 2D, extending the concept of bound states in the continuum.

Findings

Eigenstates are normalizable despite steep expulsive potentials.

Analytical asymptotic expressions match numerical solutions.

Exact vortex solutions are derived in 2D.

Abstract

On the contrary to the common intuition, which suggests that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schroedinger equations, which include expulsive potentials that are steeper than the quadratic ones, give rise to normalizable eigenstates, which may be considered as a manifestation of effective self-trapping in the linear system. These states constitute full continuous spectra in both the 1D and 2D cases. In 1D, they are spatially even and odd eigenstates. The 2D states may carry any value of the vorticity (alias magnetic quantum number). Asymptotic expressions for wave functions of the 1D and 2D eigenstates, valid far from the center, are derived analytically, demonstrating excellent agreement with full numerical solutions. Special exact solutions for vortex states are obtained in the 2D case. These findings suggest an extension of the concept of bound states in the continuum, in quantum mechanics and paraxial photonics. Gross-Pitaevskii equations are briefly considered as the nonlinear extension of the 1D and 2D settings. In 1D, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability