Sturm-Liouville operators with periodically modulated parameters. Part I: Regular case

2025/07/16 by Świderski, Grzegorz, Trojan, Bartosz
#34B24 #34C11 (Secondary) #34L05 (Primary) 34L20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2507.12300

Abstract

We introduce a new class of Sturm-Liouville operators with periodically modulated parameters. Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed for the spectral parameter equal to 0. Under certain assumptions, by studying the asymptotic behavior of Christoffel functions and density of states, we prove that the spectral density is a continuous positive everywhere function on the real line.

Citations

• Modified Jost solutions of Schrödinger operators with locally H^-1 potentials
• Asymptotic zeros' distribution of orthogonal polynomials with unbounded recurrence coefficients
• Spectral properties of Schrödinger operators with locally H^-1 potentials
• Asymptotics for Christoffel functions associated to continuum Schrödinger operators
• Perturbations of periodic Sturm--Liouville operators
• About essential spectra of unbounded Jacobi matrices
• Periodic perturbations of unbounded Jacobi matrices II: Formulas for density
• Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

Related

• Spectral Properties of Limit-Periodic Operators
• Perturbations of periodic Sturm--Liouville operators
• Indefinite Sturm-Liouville operators with periodic coefficients
• Asymptotic behaviour of the positive spectrum for a family of periodic Sturm-Liouville problems in the transition from a definite problem to indefinite one
• About one inverse problem for the Sturm-Liouville operator