Description of Spectral Functions of Differential Operators with Arbitrary Deficiency Indices

摘要

Let L_0 be the minimal operator generated by an even-order differential expression on a semiaxis. The description of all spectral functions of the operator L_0 with the maximal deficiency indices was obtained in [1] and [2]. In this note, a formula for the characteristic functions of L_0 is presented; its form is similar to that of the famous Krein formula for resolvents. This formula (see (6)) enables one to describe all spectral functions of a differential operator with arbitrary (possibly distinct) deficiency indices by using a parameter r of Nevanlinna type. Moreover, results of the papers [1] and [2] related to the case of maximal deficiency indices are also developed in the note. Namely, we show that, in this case, all characteristic functions are described by means of a linear-fractional transformation of the parameter r; here the matrix of the coefficients of this transformation satisfies relations similar to those satisfied by the Nevanlinna matrix in the moment problem (see (15) and (19)).