Odinary differential operators of odd order with distribution coefficients

Abstract

We work with differential expressions of the form \begin{align} \tau_{2n+1} y &=(-1)^ni \{(q_{0}y^{(n+1)})^{(n)}+(q_{0}y^{(n)})^{(n+1)}\}+ \sum\limits_{k=0}^{n}(-1)^{n+k}(p^{(k)}_ky^{(n-k)})^{(n-k)} \\ &\qquad+i\sum\limits_{k=1}^{n}(-1)^{n+k+1}\{(q^{(k)}_{k}y^{(n+1-k)})^{(n-k)}+ (q^{(k)}_{k}y^{(n-k)})^{(n+1-k)}\}, \end{align} where the complex valued coefficients $p_j$ and $q_j$ are subject the following conditions: $ q_0(x) \in AC_{loc}(a,b)$, $Re \,q_0>0$, while all the other functions $$q_1(x),q_2(x),\ldots,q_{n}(x), p_0(x),p_1(x),\ldots,p_n(x)$$ belong to the space $L^1_{loc}(a,b)$. This implies that the coefficients $p^{(k)}_{k}$ and $q^{(k)}_{k}$ in the expression $\tau_{2n+1}$ are distributions of singularity order $k$. The main objective of the paper is to represent the differential expression $\tau_{2n+1}$ in the other (regularized) form which allows to define the minimal and maximal operators associated with this differential expression.