On Existence of the Resolvent and Discreteness of the Spectrum of a Class of Differential Operators of Hyperbolic Type

Abstract

The existence and compactness of the resolvent are studied in this paper. One of the main results is the criterion of discreteness of the spectrum of a hyperbolic singular differential operator. Singular differential operators, for example operators defined in an unbounded domain, in general may have not only a discrete but also a continuous spectrum. Therefore in general an arbitrary function cannot be decomposed into a series of eigenfunctions. For this reason the most important problem in the study of the spectrum in dependence of the behavior of the coefficients in the case of an unbounded domain is the discreteness of the spectrum. Spectral characteristics of singular elliptic differential operators are well-studied and the typical difficulties encountered in connection with bad behaving coefficients clarified. An extensive literature is devoted to their study and we mention [1-3]. Review of the literature shows that such questions as: 1) the existence and compactness of the resolvent, 2) the discreteness of the spectrum of hyperbolic differential operators defined in an unbounded domain are not well studied. We consider in the space L 2 (Ω) the differential operator of hyperbolic type A 0 u = u xx − u yy + a (y) u x + c (y) u with the domain D(A 0) of infinitely differentiable functions satisfying the conditions u (−π; y) = u (π; y) , u x (−π; y) = u x (π; y) and compactly supported with respect to the variable y, where Ω = {(x, y) : −π < x < π, −∞ < y < ∞}. Further, we assume that the coefficients a (y) , c (y) satisfy the conditions: i) |a(y)| ≥ δ 0 > 0, c(y) ≥ δ > 0 are continuous functions in R = (−∞; ∞). It is easy to verify that the operator A 0 admits closure in the space L 2 (Ω), which is denoted by A. We note that the operator A corresponds to the problem of propagation of the boundary regime (see [1], p. 106), i.e. the problem without initial conditions. Here the term au x describes the friction force. The question of the existence of solutions of the problem without initial conditions, in general, depends on the behavior of the coefficients a and c. For example, when a = 0, the solution does not always exist. The main results of this paper are the following theorems. 1

Estimates of singular numbers (s-numbers) for a class of degenerate elliptic operators

“…The results of this work are close to those of M.B. Muratbekov [7][8][9][10], where differential operators of mixed and hyperbolic types were investigated. In contrast to the above works, here we investigate previously unconsidered degenerate elliptic equations with an arbitrary power-law degeneracy on the degeneracy line.…”