CMMSE:Estimates of approximation numbers, nuclearity of the resolvent of a third order singular differential operator and the completeness of its root vectors

Abstract

In this paper, we consider a third order singular differential operator L w + μ w = - w ′′′ + q ( x ) w + μ w in space L 2 ( R ) originally defined on the set C 0 ∞ ( R ) , where C 0 ∞ ( R ) is the set of infinitely differentiable compactly supported functions, μ≥0. Regarding the coefficient q ( x ) , we assume that it is a continuous function in R ( - ∞ , ∞ ) and can be a growing function at infinity. The operator L allows closure in the space L 2 ( R ) and the closure also be denoted by L. In the paper, under certain restrictions on q ( x ) , in addition to the above condition, the existence of the resolvent of the operator L and the existence of the estimate ‖ - w ′′′ ‖ L 2 ( R ) + ‖ q ( x ) w ‖ L 2 ( R ) ≤ c ( ‖ L w ‖ L 2 ( R ) + ‖ w ‖ L 2 ( R ) ) (0.1) have been proved, where c>0 is a constant. Example. Let q ( x ) = e 100 | x | , then the estimate (0.1) holds. The compactness of the resolvent is proved and two-sided estimates for singular numbers ( s-numbers) are obtained. Here we note that the estimates of singular numbers ( s-numbers) show the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. In the present paper, apparently for the first time, the nuclearity of the resolvent of the third-order differential operator and completeness of its root vectors are proved in the case of an unbounded domain with a greatly growing coefficient q ( x ) at infinity.