论文标题
stahl- totik的常规性，用于连续施罗丁运营商
Stahl--Totik regularity for continuum Schrödinger operators
论文作者
论文摘要
我们基于基本频谱补体的马丁紧凑型，开发了连续施罗丁师运算符的规律性理论。 This theory is inspired by Stahl--Totik regularity for orthogonal polynomials, but requires a different approach, since Stahl--Totik regularity is formulated in terms of the potential theoretic Green function with a pole at $\infty$, logarithmic capacity, and the equilibrium measure for the support of the measure, notions which do not extend to the case of unbounded spectra. For any half-line Schrödinger operator with a bounded potential (in a locally $L^1$ sense), we prove that its essential spectrum obeys the Akhiezer--Levin condition, and moreover, that the Martin function at $\infty$ obeys the two-term asymptotic expansion $\sqrt{-z} + \ frac {a} {2 \ sqrt {-z}}} + o（\ frac 1 {\ sqrt {-z}}）$ as $ z \ to - \ infty $。该扩展的常数$ a $起着适用于Schrödinger操作员的重新归一化的罗宾常数，并进入通用不平等$ a \ le \ liminf_ {x \ to \ infty} \ frac 1x \ frac 1x \ int_0^x v（t v（t）dt $。这导致了规律性的概念，并与Dirichlet溶液的根渐近学和零计数措施有联系。我们还提出了腐烂和偏执势的应用。
We develop a theory of regularity for continuum Schrödinger operators based on the Martin compactification of the complement of the essential spectrum. This theory is inspired by Stahl--Totik regularity for orthogonal polynomials, but requires a different approach, since Stahl--Totik regularity is formulated in terms of the potential theoretic Green function with a pole at $\infty$, logarithmic capacity, and the equilibrium measure for the support of the measure, notions which do not extend to the case of unbounded spectra. For any half-line Schrödinger operator with a bounded potential (in a locally $L^1$ sense), we prove that its essential spectrum obeys the Akhiezer--Levin condition, and moreover, that the Martin function at $\infty$ obeys the two-term asymptotic expansion $\sqrt{-z} + \frac{a}{2\sqrt{-z}} + o(\frac 1{\sqrt{-z}})$ as $z \to -\infty$. The constant $a$ in that expansion plays the role of a renormalized Robin constant suited for Schrödinger operators and enters a universal inequality $a \le \liminf_{x\to\infty} \frac 1x \int_0^x V(t)dt$. This leads to a notion of regularity, with connections to the root asymptotics of Dirichlet solutions and zero counting measures. We also present applications to decaying and ergodic potentials.
