On the deficiency indices of a singular differential operator of fourth order in the space of vector functions (Q845234)

The authors deal with linear differential operators of the fourth order of the form \(\ell(y)= -y^{(iv)}+ Q(x)y\), more precisely, with the minimal differential operator \(L_0\), generated by the above expression, where \(y(x)= (y_1(x), y_2(x))^T\), in the space \(M= L^2(0,\infty)\times L^2(0,\infty)\). The matrix \(Q(x)\) has the entries \(q_{ij}(x)\), \(i,j= 1,2\). Its characteristic roots are denoted \(\mu_1(x)\), \(\mu_2(x)\), both real since \(Q\) is assumed real and symmetric. Three distinct cases are considered, the deficiency indices being estimated. For instance, in the second case treated, under technical hypotheses on data, it is obtained that the indices are (4,4). In each case, \(|\mu_i(x)|\to\infty\) as \(x\to\infty\), \(i= 1,2\).