On Deficiency Indices of Singular Operators of Odd Order

Abstract

One of the main problems of the spectral theory of linear ordinary differential operators is the study of their deficiency indices depending on the behavior of the coefficients of the corresponding differential expression ly . Such problems were investigated by Naimark [1], Fedoryuk [2], Sultanaev, and others (see the bibliography in [2]). As a rule, symmetric operators with real coefficients were studied. For the case of complex coefficients, Fedoryuk obtained asymptotic formulas for the fundamental system of solutions of the equation ly = 0 under the condition that the coefficients of the differential expression ly are polynomials subject to a rigid constraint on their degree. The paper [3] was devoted to the study of a symmetric differential operator of (2n+1)th order with complex-valued coefficients of the odd powers. Classes of differential operators with deficiency indices (m, m) , where 1 ≤ m ≤ 2n − 1 , were described. The papers [4, 5] were also devoted to the study of symmetric operators in the case of complex coefficients under regularity conditions of Titchmarsh– Levitan type; asymptotic formulas for the fundamental system of solutions of the equation ly = λy were obtained, and in certain special cases the deficiency indices of the corresponding minimum operator were found. It is well known that deficiency indices define the dimension of the solution spaces of the equations lu = iy and lu = −iy . The goal of the present paper is to study the deficiency indices of a class of differential operators generated on the semiaxis by a self-adjoint differential expression of odd order with complex-valued coefficients. We consider the equation