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

TL;DR

This paper introduces a method to create time-dependent quantum potentials that produce bound states in the continuum which freeze after a certain time, using supersymmetric quantum mechanics.

Contribution

It presents a novel construction of time-dependent potentials with frozen bound states in the continuum, extending the concept to multiple states and providing explicit formulas.

Findings

Bound states in the continuum can be made to freeze after a threshold time.

The method uses supersymmetric quantum mechanics to generate these potentials.

Explicit solutions are provided for free particle initial conditions.

Abstract

In this work, we construct time-dependent potentials for the Schr\"odinger equation via supersymmetric quantum mechanics. The generated potentials have a quantum state with the property that after a particular threshold time $t_F$, when the potential does no longer change, the evolving state becomes a bound state in the continuum, its probability distribution freezes. After the factorization of a geometric phase, the state satisfies a stationary Schr\"odinger 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.