Regularization of binomial differential equations with singular coefficients

Abstract

We propose a regularization of the formal differential expression of order $m \geqslant 3$ $$ l(y) = i^my^{(m)}(t) + q(t)y(t), \,t \in (a, b), $$ applying quasi-derivatives. The distribution coefficient $q$ is supposed to have an antiderivative $Q \in L([a,b];\mathbb{C})$. For the symmetric case ($Q = \bar{Q}$) self-adjoint and maximal dissipative extensions of the minimal operator and its generalized resolvents are described. The resolvent approximation with resrect to the norm of the considered operators is also investigated. The case $m = 2$ for $Q \in L_2([a, b];\mathbb{C})$ was investigated earlier.