CMMSE: Estimates of singular numbers ( s -numbers) and eigenvalues of a
mixed elliptic-hyperbolic type operator with parabolic degeneration

## Abstract

*a*) | k ( w ) | ≥ 0 is a piecewise continuous
and bounded function in R = ( - ∞ , ∞ ) . The coefficients b ( w ) and q
( w ) are continuous functions in R and can be unbounded at infinity.
The operator *L* admits closure in the space L 2 ( Ω ) and the
closure is also denoted by *L*. Taking into consideration certain
constraints on the coefficients b ( w ) q ( w ) , apart from the
above-mentioned conditions, the existence of a bounded inverse operator
is proved in this paper; a condition guaranteeing compactness of the
resolvent kernel is found; and we also obtained two-sided estimates for
singular numbers ( *s*-numbers). Here we note that the estimate of
singular numbers ( *s*-numbers) shows the rate of approximation of
the resolvent of the operator *L* by linear finite-dimensional
operators. It is given an example of how the obtained estimates for the
*s*-numbers enable to identify the estimates for the eigenvalues of
the operator *L*. We note that the above results are apparently
obtained for the first time for a mixed-type operator in the case of an
unbounded domain with rapidly oscillating and greatly growing
coefficients at infinity.