Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients


September 14, 2018 | 2:30 - 3:30 p.m.


Campbell Hall 443


Rudi Weikard, Chair and Professor, Department of Mathematics, University of Alabama at Birmingham


We discuss the spectral theory of the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. We do not require the definiteness condition customarily made on the coefficients of the equation.

Specifically, we construct associated minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we construct Green's function and prove the existence of a spectral (or generalized Fourier) transformation.