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.