arXiv 1607.005402016-07math-phmath.MPmath.SP

Smilansky哈密顿量的共振态和束缚态

On resonances and bound states of Smilansky Hamiltonian

Pavel Exner, Vladimir Lotoreichik, Miloš Tater

摘要：

我们认为自伴Smilansky哈密顿$\mathsf{H}_\varepsilon$在$L^2(\mathbb{R}^2)$与正式的微分表达式$-\partial_x^2 - \frac12\big(\partial_y^2 + y^2) - \sqrt{2}\varepsilon y δ(x)$在亚临界制度，$\varepsilon \in (0,1)$。我们证明了存在的共振$\mathsf{H}_\varepsilon$上的一个可数的子家庭的床单的基础黎曼曲面的距离是有限的物理片。在这样的片，我们发现共振自由区和小$\varepsilon > 0$的双共振。此外，我们完善了以前已知的结果在弱耦合制度（$\varepsilon\rightarrow 0+$）的束缚态的$\mathsf{H}_\varepsilon$。在证明中，我们使用Birman-Schwinger原理的$\mathsf{H}_\varepsilon$，谱理论的Jacobi矩阵的元素，和解析隐函数定理。

英文摘要：

We consider the self-adjoint Smilansky Hamiltonian $\mathsf{H}_\varepsilon$ in $L^2(\mathbb{R}^2)$ associated with the formal differential expression $-\partial_x^2 - \frac12\big(\partial_y^2 + y^2) - \sqrt{2}\varepsilon y δ(x)$ in the sub-critical regime, $\varepsilon \in (0,1)$. We demonstrate the existence of resonances for $\mathsf{H}_\varepsilon$ on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterise resonances for small $\varepsilon > 0$. In addition, we refine the previously known results on the bound states of $\mathsf{H}_\varepsilon$ in the weak coupling regime ($\varepsilon\rightarrow 0+$). In the proofs we use Birman-Schwinger principle for $\mathsf{H}_\varepsilon$, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

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摘要：

英文摘要：

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