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Session S30 - Mathematical Methods in Quantum Mechanics

Monday, July 19, 16:30 ~ 16:55 UTC-3

Stahl-Totik regularity for continuum Schrödinger operators

Milivoje Lukić

Rice University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak8d32c76f58b1090fea7a55b99cc6639a').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy8d32c76f58b1090fea7a55b99cc6639a = 'm&#105;l&#105;v&#111;j&#101;.l&#117;k&#105;c' + '&#64;'; addy8d32c76f58b1090fea7a55b99cc6639a = addy8d32c76f58b1090fea7a55b99cc6639a + 'r&#105;c&#101;' + '&#46;' + '&#101;d&#117;'; var addy_text8d32c76f58b1090fea7a55b99cc6639a = 'm&#105;l&#105;v&#111;j&#101;.l&#117;k&#105;c' + '&#64;' + 'r&#105;c&#101;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak8d32c76f58b1090fea7a55b99cc6639a').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy8d32c76f58b1090fea7a55b99cc6639a + '\'>'+addy_text8d32c76f58b1090fea7a55b99cc6639a+'<\/a>';

We present a theory of regularity for one-dimensional continuum Schrödinger operators. For any half-line Schrödinger operator with a bounded potential $V$, we obtain universal thickness statements for the essential spectrum, in the language of potential theory and Martin functions. Namely, we prove that the essential spectrum is not polar, it obeys the Akhiezer-Levin condition, and moreover, the Martin function at infinity 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 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 limiting eigenvalue distributions for finite restrictions of the operator, and applications to decaying and ergodic potentials.

Joint work with Benjamin Eichinger (Johannes Kepler University Linz, Austria).