论文标题
带有预定义光谱间隙的周期性量子图
Periodic quantum graphs with predefined spectral gaps
论文作者
论文摘要
令$γ$为任意的$ \ mathbb {z}^n $ - 周期性公制图，它与一条线不一致。我们考虑使用动作$ - \ varepsilon^{ - 1} {\ mathrm {d}^2/\ mathrm {d}^2/\ mathrm {d} x^2} $，在$ - \ varepsilon^{ - 1} {\ varepsilon^{ - 1} {\ varepsilon^{ - 1} X^2} $上;这里$ \ varepsilon> 0 $是一个小参数。令$ m \ in \ mathbb {n} $。我们表明，在适当的顶点条件下，$ \ Mathcal {h}^\ varepsilon $的频谱$σ（\ Mathcal {h}^\ varepsilon）$至少具有$ \ varepsilon $的$ \ varepsilon $足够小。我们证明，这些间隙的渐近行为以及$σ（\ Mathcal {h}^\ varepsilon）$的底部的渐近行为作为$ \ varepsilon \ to 0 $，可以通过在这些顶点条件下的c耦合常数的合适选择来完全控制。我们还展示了如何确保固定的（足够小）$ \ varepsilon $具有带有预定义数字的第一个$ m $频谱差距的左端点的精确巧合。
Let $Γ$ be an arbitrary $\mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $\mathcal{H}_\varepsilon$ on $Γ$ with the action $-\varepsilon^{-1}{\mathrm{d}^2/\mathrm{d} x^2}$ on its edges; here $\varepsilon>0$ is a small parameter. Let $m\in\mathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $σ(\mathcal{H}^\varepsilon)$ of $\mathcal{H}^\varepsilon$ has at least $m$ gaps as $\varepsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $σ(\mathcal{H}^\varepsilon)$ as $\varepsilon\to 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $\varepsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.
