Freezable bound states in the continuum for time-dependent quantum potentials

Abstract

In this work, we construct time-dependent potentials for the Schrödinger equation via supersymmetric quantum mechanics. The Schrödinger equations with the generated potentials have a solution with the property that after a particular threshold time tF, when the potentials do no longer change, the evolving solution becomes a bound state in the continuum, its probability distribution freezes. After the factorization of a geometric phase, the solution satisfies a stationary Schrödinger equation with time-independent potential. The procedure can be extended to support more than one bound state in the continuum. Closed expressions for the potential, the bound states in the continuum, and scattering states are given for the examples starting from the free particle.