On the spectral theory for first-order systems without the unique continuation property

Abstract

We consider the differential equation $J{u}^{\prime }+qu=wf$ 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. In this situation it may happen that there is no existence and uniqueness theorem for balanced solutions of a given initial value problem. We describe the set of solutions the equation does have and establish that the adjoint of the minimal operator is still the maximal operator, even though unique continuation of balanced solutions fails.

References in corpus (1)

• Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

Cited by in corpus (1)

• Green's functions for first-order systems of ordinary differential equations without the unique continuation property