论文标题
schrödinger运算符，具有倾斜传输条件的$ \ mathbb {r}^2 $
Schrödinger operators with oblique transmission conditions in $\mathbb{R}^2$
论文作者
论文摘要
在本文中，我们在$ l^2（\ mathbb {r}^2）中研究了自我接合Schrödinger运算符的频谱，并沿着平滑的封闭曲线$σ\ subseteq \ subseteq \ mathbb {r}^2 $沿着新型的传输条件进行了新型的传输条件。尽管这些$ \ textit {斜} $传输条件与$σ$上的$δ'$ - 条件（而不是普通导数）（在这里使用的是固定器衍生物）的条件显着不同：事实证明，对于有吸引力的交互强度，离散的光谱始终从下面无基础。除了这种意外的光谱效应外，我们还确定了必需的光谱，我们证明了Krein型的分解公式和Birman-Schinginger原理。此外，我们表明，具有倾斜传播条件的这些Schrödinger算子自然出现，作为具有静电和Lorentz标量$δ$互相的狄拉克运算符的非相关限制，证明其作为量子力学模型的用法是合理的。
In this paper we study the spectrum of self-adjoint Schrödinger operators in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a smooth closed curve $Σ\subseteq \mathbb{R}^2$. Although these $\textit{oblique}$ transmission conditions are formally similar to $δ'$-conditions on $Σ$ (instead of the normal derivative here the Wirtinger derivative is used) the spectral properties are significantly different: it turns out that for attractive interaction strengths the discrete spectrum is always unbounded from below. Besides this unexpected spectral effect we also identify the essential spectrum, and we prove a Krein-type resolvent formula and a Birman-Schwinger principle. Furthermore, we show that these Schrödinger operators with oblique transmission conditions arise naturally as non-relativistic limits of Dirac operators with electrostatic and Lorentz scalar $δ$-interactions justifying their usage as models in quantum mechanics.
