Smilansky模型的正则模拟：谱性质
A regular analogue of the Smilansky model: spectral properties
摘要：
本文分析了算子$H=\frac{\partial^2}{\partial x^2} -\frac{\partial^2}{\partial y^2} +ω^2y^2-λy^2V(x y)$在$L^2(\mathbb{R}^2)$中的谱性质，其中$ω\ne 0$和$V\ge 0$是一个紧支撑的充分正则势.已知$H$的谱依赖于一维薛定谔算子$L=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ω^2-λV(x)$，并且随着$\infσ(L)$切换符号而显著改变。我们证明，在临界情况下，$\infσ(L)=0$，$H$的频谱是纯粹必要的，并涵盖区间$[0,\infty)$。在亚临界情况下，$\infσ(L)>0$，基本谱从$ω$开始，在间隔$[0,ω)$中存在非空离散谱。我们还推导出相应的特征值矩的一个界。
英文摘要：
We analyze spectral properties of the operator $H=\frac{\partial^2}{\partial x^2} -\frac{\partial^2}{\partial y^2} +ω^2y^2-λy^2V(x y)$ in $L^2(\mathbb{R}^2)$, where $ω\ne 0$ and $V\ge 0$ is a compactly supported and sufficiently regular potential. It is known that the spectrum of $H$ depends on the one-dimensional Schrödinger operator $L=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ω^2-λV(x)$ and it changes substantially as $\infσ(L)$ switches sign. We prove that in the critical case, $\infσ(L)=0$, the spectrum of $H$ is purely essential and covers the interval $[0,\infty)$. In the subcritical case, $\infσ(L)>0$, the essential spectrum starts from $ω$ and there is a non-void discrete spectrum in the interval $[0,ω)$. We also derive a bound on the corresponding eigenvalue moments.
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