Stahl--Totik regularity for continuum Schr\"odinger operators

Abstract: This talk describes joint work with Benjamin Eichinger: a theory of regularity for one-dimensional continuum Schr\"odinger operators, based on the Martin compactification of the complement of the essential spectrum. For a half-line Schr\"odinger operator $-\partial_x^2+V$ with a bounded potential $V$, it was previously known that the spectrum can have zero Lebesgue measure and even zero Hausdorff dimension; however, we obtain universal thickness statements in the language of potential theory. Namely, we prove that the essential spectrum is not polar, it obeys the Akhiezer--Levin condition, and moreover, 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 its asymptotic expansion plays the role of a renormalized Robin constant suited for Schr\"odinger 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 exponential growth rate of Dirichlet solutions and the zero counting measures for finite restrictions of the operator. We also present applications to decaying and ergodic potentials.

mathematical physics

Audience: researchers in the topic

TAMU: Mathematical Physics and Harmonic Analysis Seminar