We study the eigenpairs of a model Schrödinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line which makes the angle $\theta$ with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit $\theta \to 0$ and we prove a two term asymptotics for these eigenvalues and their associated eigenvectors: For the $n$-th eigenvalue we prove $$ \sigma_n(\theta) = \Theta_0 + (2n-1) a_1\theta + \mathcal{O}(\theta^{3/2}) $$ where $\Theta_0$ and $a_1$ are explicit quantities related to the De Gennes operator of degree $k=1$. Approximate value for these constants are $$ \eqalign {\Theta_0 \!\! &\simeq 0.590106125 \cr a_1 \!\! &\simeq 0.7651881.} $$ We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach.
19 octobre 2010
Prépublication IRMAR 10-64
Zeitschrift für angewandte Mathematik und Physik (ZAMP) Volume 63, Issue 2 (2012), Page 203-231. doi:10.1007/s00033-011-0163-y
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