Two-Term Differential Equations with Matrix Distributional Coefficients

Abstract

We propose a regularization of the formal differential expression $$\begin{array}{cc}\hfill l(y)={i}^m{y}^{(m)}(t)+q(t)y(t),\hfill & \hfill t\in \left(a,b\right)\hfill \end{array},$$ of order $m ≥ 2$ with matrix distribution $q$. It is assumed that $q = Q^{([m/2])}$, where $Q = (Q_{i,j})_{i,j = 1}^s$ is a matrix function with entries $Q_{i,j} ϵ L_2[a, b]$ if $m$ is even and $Q_{i,j} ϵ L_1[a, b]$, otherwise. In the case of a Hermitian matrix $q$, we describe self-adjoint, maximal dissipative, and maximal accumulative extensions of the associated minimal operator and its generalized resolvents.