Separability, estimates for eigenvalues and singular numbers ( s
-numbers) of a class of hyperbolic type differential operators

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 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.