2020/02/06 by Vadim Mogilevskii, Mogilevskii, Vadim
Mathematics · #34B09 #34B40 #34L10 #47A06 #47E05 #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2002.02502
openalex publication_date 2020/02/06 · openalex created_date 2022/09/16 · openalex updated_date 2026/07/28
We study pseudospectral and spectral functions for Hamiltonian system Jy'-B(t)=λΔ(t)y and differential equation l[y]=λΔ(t)y with matrix-valued coefficients defined on an interval I=[a,b) with the regular endpoint a. It is not assumed that the matrix weight Δ(t)≥ 0 is invertible a.e. on I. In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization σ=στ of all pseudospectral and spectral functions σ by means of a Nevanlinna parameter τ and single out in terms of τ and boundary conditions the class of functions y for which the inverse Fourier transform y(t)=∫ℝ φ(t,s) dσ(s) \widehat y(s) converges uniformly. We also show that for scalar equation l[y]=λΔ(t)y the set of spectral functions is not empty. This enables us to extend the Kats-Krein and Atkinson results for scalar Sturm - Liouville equation -(p(t)y')'+q(t)y=λΔ(t) y to such equations with arbitrary coefficients p(t) and q(t) and arbitrary non trivial weight Δ(t)≥ 0.