Аннотация
In this paper we study the hyperbolic operator L u + μ u = u xx - u ww + b ( w ) u x + q ( w ) u + μ u initially defined on C 0 , π ∞ ( Ω ‾ ) , where Ω ‾ = { ( x , w ) : - π ≤ x ≤ π , - ∞ < w < ∞ } , μ ≥ 0 . C 0 , π ∞ ( Ω ‾ ) is a set of infinitely differentiable functions with compact support with respect to the variable w and satisfying conditions with respect to the variable x : u ( - π , w ) = u ( π , w ) , u x ( - π , w ) = u x ( π , w ) , - ∞ < w < ∞ . With respect to the coefficients b ( w ), q ( w ) we assume that they are continuous functions in R=(-∞ , ∞) and can be strongly increasing functions at infinity. The operator L admits closure in L 2 ( Ω ) and the closure we also denote by L . In the paper, under some restrictions on the coefficients, in addition to the above conditions, we proved that there is a bounded inverse operator and found conditions on b ( w ) and q ( w ) that ensure the existence of the estimate, i.e. separability of L ‖ u xx - u ww ‖ L 2 ( Ω ) + ‖ u w ‖ L 2 ( Ω ) + ‖ b ( w ) u x ‖ L 2 ( Ω ) + ‖ q ( w ) u ‖ L 2 ( Ω ) ≤ c ⋅ ( ‖ L u ‖ L 2 ( Ω ) + ‖ u ‖ L 2 ( Ω ) ) , where c> 0 is a constant. Example 0.1. Let b ( w ) = e 1000 | w | , q ( w ) = e 100 | w | . Then the above estimate holds. In addition to the above results, the paper proves the compactness of the resolvent, obtains two-sided estimates for singular numbers ( s -numbers). Here we note that estimates of singular numbers ( s -numbers) show the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. An example is given of how these estimates allow one to find estimates for the eigenvalues of the operator under study.