Regularization of two-term differential equations with singular coefficients by quasiderivatives

Abstract

We propose a regularization of the formal differential expression $$l(y) = i^m y^{(m)}(t) + q(t)y(t),\; t \in (a, b),$$ of order $m \geq 3$ by using quasiderivatives. It is assumed that the distribution coefficient $q$ has an antiderivative $Q \in L ([a, b]; \mathbb{C})$. In the symmetric case $(Q = \overline{Q})$, we describe self-adjoint and maximal dissipative/accumulative extensions of the minimal operator and its generalized resolvents. In the general (nonselfadjoint) case, we establish conditions for the convergence of the resolvents of the considered operators in norm. The case where $m = 2$ and $Q \in L_2 ([a, b]; \mathbb{C})$ was studied earlier.