Observability Inequalities for Degenerate Schrödinger Equations via a Shape Approximation Method
Abstract
We study a two-dimensional Schrödinger equation with a boundary degeneracy. More precisely, the diffusion matrix is given by $ A=\operatorname{diag}(1,x_2^α), α\in(0,1)$, so that the partial differential operator degenerates on a portion of the boundary. The main objective is to establish a boundary observability inequality for the corresponding degenerate Schrödinger equation. To overcome the difficulties caused by the boundary degeneracy, we introduce a shape design and approximation procedure. By removing a thin neighborhood of the degenerate boundary, we construct a family of uniformly Schrödinger equations on perturbed domains. We first investigate the spectral properties of the associated operators and establish the convergence of the approximate problems to the original degenerate problem. We then derive a boundary observability inequality for the uniformly approximating equations and pass to the limit as the thickness of the removed boundary layer tends to zero. This yields a boundary observability inequality for the original degenerate Schrödinger equation. Our approach combines spectral analysis, separation of variables, and shape design techniques. By employing multiplier methods for the uniformly approximating problems and passing to the degenerate limit, we establish the boundary observability inequality for the original degenerate Schrödinger equation.
Related
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BibTeX
@misc{yang2026observability,
title = {{Observability Inequalities for Degenerate Schrödinger Equations via a Shape Approximation Method}},
author = {Dong-Hui Yang and Hongli Sun and Weijia Wu and Yuanzhi Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33406},
eprint = {2610.33406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33406},
}
RIS
TY - UNPB
TI - Observability Inequalities for Degenerate Schrödinger Equations via a Shape Approximation Method
AU - Dong-Hui Yang
AU - Hongli Sun
AU - Weijia Wu
AU - Yuanzhi Zhou
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33406
UR - https://arxiv.org/abs/2610.33406
AB - We study a two-dimensional Schrödinger equation with a boundary degeneracy. More precisely, the diffusion matrix is given by $ A=\operatorname{diag}(1,x_2^α), α\in(0,1)$, so that the partial differential operator degenerates on a portion of the boundary. The main objective is to establish a boundary observability inequality for the corresponding degenerate Schrödinger equation. To overcome the difficulties caused by the boundary degeneracy, we introduce a shape design and approximation procedure. By removing a thin neighborhood of the degenerate boundary, we construct a family of uniformly Schrödinger equations on perturbed domains. We first investigate the spectral properties of the associated operators and establish the convergence of the approximate problems to the original degenerate problem. We then derive a boundary observability inequality for the uniformly approximating equations and pass to the limit as the thickness of the removed boundary layer tends to zero. This yields a boundary observability inequality for the original degenerate Schrödinger equation. Our approach combines spectral analysis, separation of variables, and shape design techniques. By employing multiplier methods for the uniformly approximating problems and passing to the degenerate limit, we establish the boundary observability inequality for the original degenerate Schrödinger equation.
ER -