arXiv 0908.14402009-08-11math-phmath.MPmath.SP

半直线Schr dinger算子再生核的普适极限与本征值的钟行为

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

Anna Maltsev

摘要：

我们将Lubinsky，Levin，Simon和Totik的一些最近的结果从紧支集的测度推广到半直线上薛定谔算子的谱测度.特别地，我们定义了薛定谔算子的再生核$S_L$，并利用它研究了具有扰动周期势的半线薛定谔算子盒中本征值的精细间距.我们表明，如果解决方案$u(ξ, x)$有界的$x$的$e^{εx}$一致的$ξ$附近的频谱在平均意义上和频谱测量是积极的，绝对连续的有界区间$I$在内部的频谱与$ξ_0\in I$，那么一致的$I$ $$\frac{S_L(ξ_0 + a/L, ξ_0 + b/L)}{S_L(ξ_0, ξ_0)} \to \frac{\sin(πρ(ξ_0)(a - b))}{πρ(ξ_0)(a - b)},$$其中$ρ(ξ)dξ$是状态密度。我们推导出在一个尺寸为$L$的大盒子中，$ξ_0$附近的特征值的间距渐近为$\frac{1}{Lρ}$。我们适应的方法来显示类似的结果正交多项式。

英文摘要：

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel $S_L$ for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions $u(ξ, x)$ are bounded in $x$ by $e^{εx}$ uniformly for $ξ$ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval $I$ in the interior of the spectrum with $ξ_0\in I$, then uniformly in $I$ $$\frac{S_L(ξ_0 + a/L, ξ_0 + b/L)}{S_L(ξ_0, ξ_0)} \to \frac{\sin(πρ(ξ_0)(a - b))}{πρ(ξ_0)(a - b)},$$ where $ρ(ξ)dξ$ is the density of states. We deduce that the eigenvalues near $ξ_0$ in a large box of size $L$ are spaced asymptotically as $\frac{1}{Lρ}$. We adapt the methods used to show similar results for orthogonal polynomials.

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