On the stability of degenerate Schr\"{o}dinger equation under boundary fractional damping

TL;DR

This paper investigates the well-posedness and stability of a degenerate Schrödinger equation with fractional boundary damping, demonstrating exponential and polynomial decay rates of solutions.

Contribution

It establishes the well-posedness and decay rates for a degenerate Schrödinger equation with boundary fractional damping, a novel analysis in this context.

Findings

Proved well-posedness of the degenerate Schrödinger problem.

Established exponential decay of solutions.

Demonstrated polynomial decay under certain conditions.

Abstract

In this paper we study the well-posedness and stability of degenerate Schr\"{o}dinger equation with a fractional boundary damping. First, we establish the well-posedness of the degenerate problem $\psi_t(x,t)-\imath(\tau(x) \psi_x(x,t))_x=0, \hbox{ with } x \in (0,1)$, controlled by Dirichlet-Neumann conditions. Then, exponential and polynomial decay rate of the solution are established using multiplier method.