ABSTRACT

We investigate the Schrödinger operators $H_{ϵ}=-\mathrm{\Delta }+W+V_{ϵ}$ in ${\mathbb{R}}^2$ with the short-range potentials $V_{ϵ}$ which are localized around a smooth closed curve γ. The operators $H_{ϵ}$ can be viewed as an approximation of the heuristic Hamiltonian $H=-\mathrm{\Delta }+W+a{\mathrm{\partial }}_{\nu }{\delta }_{\gamma }+b{\delta }_{\gamma }$, where ${\delta }_{\gamma }$ is Dirac's δ-function supported on γ and ${\mathrm{\partial }}_{\nu }{\delta }_{\gamma }$ is its normal derivative on γ. Assuming that the operator $-\mathrm{\Delta }+W$ has only discrete spectrum, we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of $H_{ϵ}$. The transmission conditions on γ for the eigenfunctions $u^{+}=\theta u^{-}$, $\theta {\mathrm{\partial }}_{\nu }{u}^{+}-{\mathrm{\partial }}_{\nu }{u}^{-}=\text{β}{u}^{-}$ which arise in the limit as $ϵ\to 0$ reveal a nontrivial connection between spectral properties of $H_{ϵ}$ and the geometry of γ.

摘要

我们研究薛定谔算子$H_{\epsilon }=-\mathrm{\Delta }+W+{五}_{\epsilon }$在${\mathbb{R}}^2$具有短程潜力${五}_{\epsilon }$它们位于平滑的闭合曲线γ周围。运营商$H_{\epsilon }$可以看作是启发式哈密顿量的近似$H=-\mathrm{\Delta }+W+\mathrm{一个}{\mathrm{\partial }}_{\nu }{\delta }_{\gamma }+b{\delta }_{\gamma }$， 在哪里${\delta }_{\gamma }$是γ支持的狄拉克δ函数和${\mathrm{\partial }}_{\nu }{\delta }_{\gamma }$是它对γ的正规导数。假设运营商$-\mathrm{\Delta }+W$只有离散谱，我们分析特征值和特征函数的渐近行为$H_{\epsilon }$. 本征函数在γ上的传输条件${你}^{+}=\theta {你}^{-}$,$\theta {\mathrm{\partial }}_{\nu }{你}^{+}-{\mathrm{\partial }}_{\nu }{你}^{-}=\text{β}{你}^{-}$在极限中出现$\epsilon \to 0$揭示了光谱特性之间的非平凡联系$H_{\epsilon }$和γ的几何形状。