Continuations of Hermitian indefinite functions and corresponding canonical systems: an example

Heinz Langer, Matthias Langer, Zoltán Sasvári

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Abstract

M. G. Krein established a close connection between the continuation problem of positive definite functions from a finite interval to the real axis and the inverse spectral problem for differential operators. In this note we study such a connection for the function f(t) = 1 − |t|, t - R, which is not positive definite on R: its restrictions fa := f|(−2a,2a) are positive definite if a ≤ 1 and have one negative square if a > 1. We show that with f a canonical differential equation or a Sturm-Liouville equation can be associated which have a singularity.
Original languageEnglish
Pages (from-to)39-53
Number of pages15
JournalMethods of Functional Analysis and Topology
Volume10
Issue number1
Publication statusPublished - 2004

Keywords

  • Hermitian indefinite functions
  • canonical systems
  • Sturm-Liouville equation

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