Strong coupling asymptotics for $\delta$-interactions supported by curves with cusps
| Autor: | Flamencourt, Brice, Pankrashkin, Konstantin |
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| Rok vydání: | 2019 |
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| Zdroj: | J. Math. Anal. Appl. 491 (2020) 124287 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.jmaa.2020.124287 |
| Popis: | Let $\Gamma\subset \mathbb{R}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues of the Schr\"odinger operator $H_\alpha$ with the attractive $\delta$-potential of strength $\alpha>0$ supported by $\Gamma$, which is defined by its quadratic form \[ H^1(\mathbb{R}^2)\ni u\mapsto \iint_{\mathbb{R}^2} |\nabla u|^2\,\mathrm{d}x-\alpha\int_\Gamma u^2\, \mathrm{d}s, \] where $\mathrm{d}s$ stands for the one-dimensional Hausdorff measure on $\Gamma$. It is shown that if $n\in\mathbb{N}$ is fixed and $\alpha$ is large, then the well-defined $n$th eigenvalue $E_n(H_\alpha)$ of $H_\alpha$ behaves as \[ E_n(H_\alpha)=-\alpha^2 + 2^{\frac{2}{p+2}} \mathcal{E}_n \,\alpha^{\frac{6}{p+2}} + \mathcal{O}(\alpha^{\frac{6}{p+2}-\eta}), \] where the constants $\mathcal{E}_n>0$ are the eigenvalues of an explicitly given one-dimensional Schr\"odinger operator determined by the cusp, and $\eta>0$. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when~$\Gamma$ is smooth or piecewise smooth with non-zero angles. Comment: 27 pages |
| Databáze: | arXiv |
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