Stability analysis for a coupled Schrödinger system with one boundary damping

• Published in: Mathematical methods in the applied sciences 2023-09, Vol.46 (14), p.14771-14793
• Main Author: Zhang, Hua‐Lei
• Format: Article
• Language: English
• Subjects: Asymptotic series; Boundary damping; Decay rate; Eigenvalues; Eigenvectors; Frequency domain analysis; Mathematical analysis; Schrodinger equation; Schrödinger system; stability; Stability analysis
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9344

In this paper, we study the stability of a Schrödinger system with one boundary damping, which consists of two constant coefficients Schrödinger equations coupled through zero‐order terms. First, we show that when ϱ=1,$$ \varrho =1, $$ the one‐dimensional Schrödinger system is not exponentially stable by the asymptotic expansions of eigenvalues. Then, by the frequency domain approach and the multiplier method, we show that the energy decay rate of the multidimensional Schrödinger system is t−1$$ {t}^{-1} $$ for sufficiently smooth initial data when ϱ=1,$$ \varrho =1, $$ |α|$$ \mid \alpha \mid $$ is sufficiently small, and the boundary of domain satisfies suitable geometric assumption. Next, by solving the characteristic equation of unbounded operator, we show that the strong stability of the one‐dimensional Schrödinger system is completely determined by ϱ$$ \varrho $$ and α$$ \alpha $$ and give the necessary and sufficient condition that ϱ$$ \varrho $$ and α$$ \alpha $$ satisfy. Finally, by solving the resolvent equation of unbounded operator and using the frequency domain approach, we show that when ϱ≠1$$ \varrho \ne 1 $$ and |α|$$ \mid \alpha \mid $$ is small enough, the energy of the one‐dimensional Schrödinger system decays polynomially and the decay rate depends on the arithmetic property of ϱ.$$ \varrho . $$