Stability and control of a 1D quantum system with confining time dependent delta potentials

TL;DR

This paper studies the stability and controllability of a one-dimensional quantum system with a time-dependent point interaction, proving existence of solutions and analyzing control via interaction strength.

Contribution

It establishes the existence of solutions for the Schrödinger equation with time-dependent delta potentials and explores control and stability properties of the quantum system.

Findings

Proved existence of strict solutions under regularity assumptions.

Analyzed stability of the quantum state.

Demonstrated local controllability using interaction strength.

Abstract

The evolution problem for a quantum particle confined in a 1D box and interacting with one fixed point through a time dependent point interaction is considered. Under suitable assumptions of regularity for the time profile of the Hamiltonian, we prove the existence of strict solutions to the corresponding Schr\"odinger equation. The result is used to discuss the stability and the steady-state local controllability of the wavefunction when the strenght of the interaction is used as a control parameter.