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

Langer, H. and Langer, M. and Sasvari, Z. (2004) Continuations of Hermitian indefinite functions and corresponding canonical systems: an example. Methods of Functional Analysis and Topology, 10 (1). pp. 39-53. ISSN 1029-3531

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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.