具有非局部扰动的线性周期薛定谔算子的局部模
Localized Modes of the Linear Periodic Schrödinger Operator with a Nonlocal Perturbation
摘要：
我们考虑了具有界面的周期薛定谔算符$-\partial_x^2+ V(x)$的本征值所对应的局域模的存在性。该接口是由跳跃的值或衍生物的$V(x)$，在一般情况下，不对应于一个本地化的扰动的完美的周期性运营商。此外，在界面的每一侧上的周期性电势可以是不同的。正如我们所示，本征值只能出现在谱隙中。我们提出的本征值问题作为一个$C^1$胶合问题的基本解（布洛赫函数）的二阶常微分方程的每一边的界面。因此，该问题被简化为找到比率函数$R_\pm=\frac{ψ_\pm'(0)}{ψ_\pm(0)}$的匹配，其中$ψ_\pm$是在相应的半线上衰减的布洛赫函数。这些比率函数的分析与Prüfer变换的帮助下。带边$R_\pm$的极限值取决于带边Dirichlet和Neumann本征值的排序。我们表明，通过变分分析的潜力，满足一定的单调性条件，可以确定在前两个间隙的顺序。通过界面特征值的数值计算验证了分析结果。
英文摘要：
We consider the existence of localized modes corresponding to eigenvalues of the periodic Schrödinger operator $-\partial_x^2+ V(x)$ with an interface. The interface is modeled by a jump either in the value or the derivative of $V(x)$ and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be different. As we show, eigenvalues can only occur in spectral gaps. We pose the eigenvalue problem as a $C^1$ gluing problem for the fundamental solutions (Bloch functions) of the second order ODEs on each side of the interface. The problem is thus reduced to finding matchings of the ratio functions $R_\pm=\frac{ψ_\pm'(0)}{ψ_\pm(0)}$, where $ψ_\pm$ are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Prüfer transformation. The limit values of $R_\pm$ at band edges depend on the ordering of Dirichlet and Neumann eigenvalues at gap edges. We show that the ordering can be determined in the first two gaps via variational analysis for potentials satisfying certain monotonicity conditions. Numerical computations of interface eigenvalues are presented to corroborate the analysis.
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