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

Vadim Mogilevskii

Mathematische Nachrichten, 2022, vol. 295, issue 9, 1755-1782

Abstract: We study pseudospectral and spectral functions for Hamiltonian system Jy′−B(t)=λΔ(t)y$Jy^{\prime }-B(t)=\lambda \Delta (t)y$ and differential equation l[y]=λΔ(t)y$l[y]=\lambda \Delta (t)y$ with matrix‐valued coefficients defined on an interval I=[a,b)$\mathcal {I}=[a,b)$ with the regular endpoint a. It is not assumed that the matrix weight Δ(t)≥0$\Delta (t)\ge 0$ is invertible a.e. on I$\mathcal {I}$. In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization σ=στ$\sigma =\sigma _\tau$ 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)=∫Rφ(t,s)dσ(s)ŷ(s)$y(t)=\int _\mathbb {R}\varphi (t,s)\, d\sigma (s) \widehat{y}(s)$ converges uniformly. We also show that for scalar equation l[y]=λΔ(t)y$l[y]=\lambda \Delta (t)y$ 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=λΔ(t)y$-(p(t)y^{\prime })^{\prime }+q(t)y=\lambda \Delta (t) y$ to such equations with arbitrary coefficients p(t)$p(t)$ and q(t)$q(t)$ and arbitrary non trivial weight Δ(t)≥0$\Delta (t)\ge 0$.

Date: 2022
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