On spectral theory for Schrödinger operators with strongly singular potentials

By Fritz Gesztesy, Maxim Zinchenko. 2005

136 citations recorded by the source corpus.

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)

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f6a3dc3f253acf170a20a0a9993b66f84db663e4

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Citations in the corpus

Boundary Value Problems, Weyl Functions, and Differential Operators (2020, background): “For singular perturbations associated with Sturm–Liouville operators, see for instance [331] and the later papers [9, 28, 29, 124, 171, 182, 282, 315, 316, 479, 480, 481, 482, 507, 531, 532, 549, 550]; for δ-point interactions we refer to [8, 293].”
Quasilimiting behavior for one-dimensional diffusions with killing (2010, background): “A proof of it can be found in general references on the theory of SturmLiouville or Schrödinger operators, such as [20, 8, 58].”
Sturm-Liouville Operators with Singularities and Generalized Nevanlinna Functions (2010, background): “As a consequence, the differential operators considered in this paper, also in the cases of two singular endpoints, have simple spectrum; compare also [14], [9], [12], and the corresponding remarks in [10].”
An Operator Theoretic Interpretation of the Generalized Titchmarsh-Weyl Coefficient for a Singular Sturm-Liouville Problem (2011, background): “For the differential expression under consideration it has been introduced in [15] and further investigated in [16], see also [17].”