On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight

Abstract

We study pseudospectral and spectral functions for Hamiltonian system $Jy'-B(t)=\lambda\Delta(t)y$ and differential equation $l[y]=\lambda\Delta(t)y$ with matrix-valued coefficients defined on an interval $\mathcal{I}=[a,b)$ with the regular endpoint $a$. It is not assumed that the matrix weight $\Delta(t)\geq 0$ is invertible a.e. on $\mathcal{I}$. In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization $\sigma=\sigma_\tau$ of all pseudospectral and spectral functions $\sigma$ by means of a Nevanlinna parameter $\tau$ and single out in terms of $\tau$ and boundary conditions the class of functions $y$ for which the inverse Fourier transform $y(t)=\int\limits_{\mathbb{R}} \varphi(t,s)\, d\sigma (s) \widehat y(s)$ converges uniformly. We also show that for scalar equation $l[y]=\lambda \Delta(t)y$ the set of spectral functions is not empty. This enables us to extend the Kats-Krein and Atkinson results for scalar Sturm - Liouville equation $-(p(t)y')'+q(t)y=\lambda \Delta (t) y$ to such equations with arbitrary coefficients $p(t)$ and $q(t)$ and arbitrary non trivial weight $\Delta (t)\geq 0$.