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On spectral theory for Schrödinger operators with strongly singular potentials
Author(s) -
Gesztesy Fritz,
Zinchenko Maxim
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510410
Subject(s) - mathematics , operator (biology) , schrödinger's cat , context (archaeology) , function (biology) , half line , point (geometry) , mathematical analysis , spectral theory , line (geometry) , pure mathematics , matrix (chemical analysis) , boundary value problem , hilbert space , geometry , paleontology , biochemistry , chemistry , materials science , repressor , evolutionary biology , biology , transcription factor , composite material , gene
We examine two kinds of spectral theoretic situations: First, we recall the case of self‐adjoint half‐line Schrödinger operators on [ a , ∞), a ∈ ℝ, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix‐valued Herglotz functions representing the associated Weyl–Titchmarsh coefficients. Second, we contrast this with the case of self‐adjoint half‐line Schrödinger operators on ( a , ∞) with a potential strongly singular at the end point a . We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is self‐adjoint (equivalently, the associated minimally defined Schrödinger operator is essentially selfadjoint) and hence no boundary condition is required at the finite end point a . For this case we show that the Weyl–Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)