论文标题
部分可观测时空混沌系统的无模型预测
Bound states of weakly deformed soft waveguides
论文作者
论文摘要
在本文中，我们考虑具有诱人潜力的二维Schrödinger操作员，这是一个无限的条状区域的特征功能的倍数$ f $是一个紧凑的连续功能。我们证明，如果$ \ int _ {\ Mathbb {r}} f \，\ Mathsf {d} x> 0 $，那么相应的schrödinger操作员在所有足够小的$ \ varepsilon> 0 $ undime of as y as y asmptiles的基本频谱下方的schrödingeroperator在基本频谱的阈值以下具有独特的简单特征值，我们可以在这一范围内获得这一范围。 $ \ varepsilon \ rightarrow 0 $。还获得了相应特征功能的渐近扩展为$ \ varepsilon \ rightarrow 0 $。如果$ \ int _ {\ mathbb {r}} f \，\ mathsf {d} x <0 $，我们证明所有足够小的$ \ varepsilon> 0 $的离散频谱都是空的。在关键情况下，$ \ int _ {\ mathbb {r}} f \，\ mathsf {d} x = 0 $，我们为所有足够小的$ \ varepsilon> 0 $ 0 $ 0 $ 0 $ 0 $ 0 $提供了足够的条件。
In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function $\mathbb{R}\ni x \mapsto d+\varepsilon f(x)$, where $d > 0$ is a constant, $\varepsilon > 0$ is a small parameter, and $f$ is a compactly supported continuous function. We prove that if $\int_{\mathbb{R}} f \,\mathsf{d} x > 0$, then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small $\varepsilon >0$ and we obtain the asymptotic expansion of this eigenvalue in the regime $\varepsilon\rightarrow 0$. An asymptotic expansion of the respective eigenfunction as $\varepsilon\rightarrow 0$ is also obtained. In the case that $\int_{\mathbb{R}} f \,\mathsf{d} x < 0$ we prove that the discrete spectrum is empty for all sufficiently small $\varepsilon > 0$. In the critical case $\int_{\mathbb{R}} f \,\mathsf{d} x = 0$, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small $\varepsilon > 0$.
