论文标题
2DSchrödinger操作员具有奇异电位的曲线附近
2D Schrödinger operators with singular potentials concentrated near curves
论文作者
论文摘要
我们研究了$ h_ \ varepsilon =-Δ+w+v_ \ varepsilon $ in $ \ mathbb {r}^2 $，带有短范围的电位$ v_ \ varepsilon $，它们本地定位在平滑的封闭曲线$γ$上。 The operators $H_\varepsilon$ can be viewed as an approximation of the heuristic Hamiltonian $H=-Δ+W+a\partial_νδ_γ+bδ_γ$, where $δ_γ$ is Dirac's $δ$-function supported on $γ$ and $\partial_νδ_γ$ is its normal derivative on $γ$.假设运算符$-δ+W $仅具有离散频谱，我们将分析$ H_ \ Varepsilon $的特征值和特征功能的渐近行为。 $γ$的传输条件$ u^+= au^ - $，$α\，\partial_νu^+ - \partial_νu^ - =βu^ - $ =βu^ - $，在$ \ varepsilon \ to 0 $中呈$ $ y $ $ y $ y y y y y y y h y y y y y y $ \ h h h y y y y y y y y y y y y y y y y y y y $ \ h h y y y y y y y y y y $ \ h y y y y y $ \ h y y y $
We investigate the Schrödinger operators $H_\varepsilon=-Δ+W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $γ$. The operators $H_\varepsilon$ can be viewed as an approximation of the heuristic Hamiltonian $H=-Δ+W+a\partial_νδ_γ+bδ_γ$, where $δ_γ$ is Dirac's $δ$-function supported on $γ$ and $\partial_νδ_γ$ is its normal derivative on $γ$. Assuming that the operator $-Δ+W$ has only discrete spectrum, we analyze the asymptotic behaviour of eigenvalues and eigenfunctions of $H_\varepsilon$. The transmission conditions on $γ$ for the eigenfunctions $u^+=αu^-$, $α\, \partial_νu^+-\partial_νu^-=βu^-$, which arise in the limit as $\varepsilon\to 0$, reveal a nontrivial connection between spectral properties of $H_\varepsilon$ and the geometry of $γ$.
