On the stability of degenerate Schrödinger equation under boundary fractional damping

TL;DR

The paper analyzes a degenerate Schrödinger equation with fractional boundary damping by embedding memory into an augmented model and formulating the problem in a weighted Hilbert space. Through semigroup theory, it proves well-posedness by showing dissipativity and surjectivity of the generator, yielding a contraction semigroup with unique solutions. It then establishes asymptotic stability via spectral analysis, proving strong stability and, under nonzero damping, a polynomial energy decay rate that is shown to be sharp. Finally, it proves the lack of exponential stability by constructing eigenvalues approaching the imaginary axis for certain degeneracy profiles, confirming that the polynomial decay rate is optimal in general.

Abstract

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

Theorems & Definitions (13)

• Proposition 2.1: see mbod
• Lemma 2.1: see achour.1
• Lemma 2.2
• Theorem 3.1
• Theorem 4.1: AB
• Theorem 4.2
• Lemma 4.1
• Lemma 4.2
• Theorem 4.3
• Theorem 4.4