H. Langer, M. Langer and Z. Sasvári,
Continuations of Hermitian indefinite functions and corresponding canonical systems: an example,
Methods Funct. Anal. Topology 10 (2004), 39–53

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.

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