论文标题
稀薄波导中二维狄拉克操作员的光谱渐近学
Spectral asymptotics for two-dimensional Dirac operators in thin waveguides
论文作者
论文摘要
我们认为在$ C^4 $ - 平面曲线的管状邻域中构成的二维DIRAC操作员。根据对曲率$κ$的一般假设，我们证明在薄宽度方案中，特征值的分裂由一维schrödinger操作员驱动于$ l^2（\ mathbb r）$ \ [ \ Mathcal {l} _e：= - \ frac {d^2} {ds^2} - \ frac {κ^2} {π^2} \]具有几何诱导的电位。特征值显示在订单$ \ varepsilon $的距离上，其中$ 2 \ varepsilon $是波导的宽度。这与该模型的非相对论对应物相反，该模型的距离为有限距离。
We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a $C^4$-planar curve. Under generic assumptions on its curvature $κ$, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schrödinger operator on $L^2(\mathbb R)$ \[ \mathcal{L}_e := -\frac{d^2}{ds^2} - \frac{κ^2}{π^2} \] with a geometrically induced potential. The eigenvalues are shown to be at distance of order $\varepsilon$ from the essential spectrum, where $2\varepsilon$ is the width of the waveguide. This is in contrast with the non-relativistic counterpart of this model, for which they are known to be at a finite distance.
