On the completeness of root functions of pencils of linear ordinary differential operators with general boundary conditions (Q1776242)

Consider pencils of the form \[ l(y)= \sum_{k_0+ k_1\leq n} \lambda^{k_0} A^{(k_0 k_1)}(x) y^{(k_1)}(x),\quad a< x< b,\quad A^{(0,n)}\equiv 1,\tag{1} \] with the boundary conditions \[ L\cdot Y(a)+ R\cdot Y(b)= 0.\tag{2} \] Here, \(Y(x):= (y(x), y'(x), y''(x),\dots, y^{(n-1)}(x))^T\), \(L\) and \(R\) are \(n\times n\)-matrices, without loss of generality \(0< l:= \text{rank}(R)\leq\text{rank}(L)\). The main results of the paper provide various sufficient conditions for the \(n\)-fold completeness of the system of root functions of the pencil (1) and (2). In each of these sufficient conditions, one considers the complex roots \(\phi\) of the equation \[ \sum^n_{j=0} A^{(jn- j)}(x)\phi^{n-j}= 0 \] and makes several purely geometrical assumptions about the location of these roots in the complex plane as well as an assumption relating the location of these roots with the rank \(l\). Moreover, various regularity assumptions on the functions \(A^{(k_0k_1)}\) are made.