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.