The continuous spectrum of bound states in expulsive potentials: self-trapping in the linear system

Introduction: Contrary to common intuition, which suggests that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schrödinger 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. Materials and methods: 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). Results: 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. Conclusions: 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 the 1D state, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability.