2020/12/31 by Branko Ćurgus, Ćurgus, Branko, Volodymyr Derkach +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #34B24 #34L10 #46C20 #47B25 #47B50 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2101.00104
openalex publication_date 2020/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider the indefinite Sturm-Liouville differential expression \mathfraka(f) := - (1)/(w)( (1)/(r) f' )', where \mathfraka is defined on a finite or infinite open interval I with 0∈ I and the coefficients r and w are locally summable and such that r(x) and (sgn x) w(x) are positive a.e. on I. With the differential expression \mathfraka we associate a nonnegative self-adjoint operator A in the Krein space L2w(I), which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of I with the positive and the negative semi-axis. For the operator A we derive conditions in terms of the coefficients w and r for the existence of a Riesz basis consisting of generalized eigenfunctions of A and for the similarity of A to a self-adjoint operator in a Hilbert space L2|w|(I). These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces, which are couplings of two symmetric operators acting in Hilbert spaces.