Schrödinger Operators with Periodic Singular Potentials

2001/09/19 by Rostyslav O. Hryniv, Hryniv, Rostyslav O., Yaroslav V. Mykytyuk +1 · 1 citation
Mathematics · Physics and Astronomy · #34L05 (Primary) 34L40 #47A10 #47B25 (Secondary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:34L05 #msc:34L40 #msc:47A10 #msc:47B25

paper · pdf · doi:10.48550/arxiv.math/0109129

12 pages, Amslatex, To appear in Meth. Funct. Anal. Topol

arxiv created 2001/09/19 · arxiv updated 2009/11/30

Abstract

We show that formal Schrödinger operators with singular potentials from the space W^-1_2,unif(R) can be naturally defined to give selfadjoint and bounded below operators, which depend continuously in the uniform resolvent sense on the potential in the W^-1_2,unif(R)-norm. In the case of periodic singular potentials we also establish pure absolute continuity and a band and gap structure of the spectrum thus generalising some classical results for singular potentials of one-dimensional quasicrystal theory.

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• A generalized Birman-Schwinger principle and applications to one-dimensional Schrödinger operators with distributional potentials

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