The Spectra of General Differential Operators in the Direct Sum Spaces

Abstract

In this paper, the general ordinary quasi-differential expression M pof n-th order with complex coefficients and its formal adjoint M +p on any finite number of intervals I p =(a p ,b p ),p= 1,...,N, are considered in the setting of the direct sums of L 2 wp (a p ,b p )-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations of those in a symmetric case in [1], [14], [15], [16], [17] and of a general case with one interval in [2], [11], [12], whilst others are new.