Matematica}2(27) Matematika FINITENESS OF EIGENVALUES OF SOME INTEGRO-DIFFERENTIAL OPERATORS

Abstract. In this note, results on finiteness of point spectrum of operators generated by integro-differential expression of arbitrary order are announced. The problem of finiteness of the point spectrum of a certain class of nonselfadjoint operators has been studied in many works [1–6]. Usually, in such cases, it is used the method of holomorphic extension of resolvent of unperturbed operator through continuous spectrum [2,6,8]. In many works the problem has been reduced to the investigation of analyticity of function (see, for example [1,4]), as well as to solving some integral equations, using Fredholm-type determinants [8]. In the present note the problem is solved by straightforward studying of analy-ticity of resolvent function. The self-adjoint case has been investigated in [7]. It should be mentioned that the case of the space L2(R+) is considered, but all results obtained here are also true in the case of the whole real axis. I. Let H be the integro-differential operator defined by H = n∑ α,β=0 D∗αMαβDβ, (1) where (Mαβu)(x) = aαβu(x)+qαβ(x)u(x)