On the deficiency numbers of ordinary differential operators

In the talk we will discuss some problems connected to the deficiency index of a minimal operator generated by an ordinary symmetric differential expression on the semiaxis in the space $L_2$. In particular, among the themes under consideration are:
1) The definition of a quasi-differential expression (a differential expression with non-smooth coefficients), of a minimal operator generated by this expression, and of its deficiency numbers.
2) The divergent form of a real differential expression with smooth coefficients and McLeod's conjecture.
3) A criterium of the maximality of the deficiency numbers of a minimal operator in terms of the Cauchy function.
4) The limit point and the limit disc for the Sturm–Liouville operator.
5) Some open problems.