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.
|Number of pages||15|
|Journal||Methods of Functional Analysis and Topology|
|Publication status||Published - 2004|
- Hermitian indefinite functions
- canonical systems
- Sturm-Liouville equation
Langer, H., Langer, M., & Sasvári, Z. (2004). Continuations of Hermitian indefinite functions and corresponding canonical systems: an example. Methods of Functional Analysis and Topology, 10(1), 39-53.