Eigenvalues of Robin Laplacians in infinite sectors
| Autor: | Konstantin Pankrashkin, Magda Khalile |
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| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
General Mathematics 010102 general mathematics Essential spectrum Eigenfunction Star (graph theory) Directional derivative 01 natural sciences Robin boundary condition Mathematics - Spectral Theory Mathematics - Analysis of PDEs 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Asymptotic expansion Spectral Theory (math.SP) Laplace operator Eigenvalues and eigenvectors Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Mathematische Nachrichten. 291:928-965 |
| ISSN: | 0025-584X |
| Popis: | For $\alpha\in(0,\pi)$, let $U_\alpha$ denote the infinite planar sector of opening $2\alpha$, \[ U_\alpha=\big\{ (x_1,x_2)\in\mathbb R^2: \big|\arg(x_1+ix_2) \big|0$. The essential spectrum of $T^\gamma_\alpha$ does not depend on the angle $\alpha$ and equals $[-\gamma^2,+\infty)$, and the discrete spectrum is non-empty iff $\alpha0$, and the $n$th eigenvalue $E_n(T^\gamma_\alpha)$ of $T^\gamma_\alpha$ behaves as \[ E_n(T^\gamma_\alpha)=-\dfrac{\gamma^2}{(2n-1)^2 \alpha^2}+O(1) \] and admits a full asymptotic expansion in powers of $\alpha^2$. The eigenfunctions are exponentially localized near the origin. The results are also applied to $\delta$-interactions on star graphs. Comment: 34 pages |
| Databáze: | OpenAIRE |
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