Estimates of singular numbers (s$$ s $$‐numbers) and eigenvalues of a mixed elliptic‐hyperbolic type operator with parabolic degeneration.

This paper is concerned with a mixed type differential operator Lu=kyuxx−uyy+byux+qyu,$$ Lu=k(y){u}_{xx}-{u}_{yy}+b(y){u}_x+q(y)u, $$which is initially defined with C0,π∞Ω‾$$ {C}_{0,\pi}^{\infty}\left(\overline{\Omega}\right) $$, where Ω‾={x,y:−π≤x≤π,−∞<y<∞}$$ \overline{\Omega}=\left\{\left(x,y\right):-\pi \le x\le \pi, -\infty <y<\infty \right\} $$ and C0,π∞$$ {C}_{0,\pi}^{\infty } $$ is a set of infinitely differentiable functions with compact support with respect to the variable y$$ y $$ and satisfying the conditions: uxi−π,y=uxiπ,yi=0,1.$$ {u}_x^{(i)}\left(-\pi, \kern3.0235pt y\right)={u}_x^{(i)}\left(\pi, \kern3.0235pt y\right)\kern0.60em i=0,\kern3.0235pt 1. $$ Regarding the coefficient ky$$ k(y) $$, with supposition that ky$$ k(y) $$ satisfies the condition: a)ky≥0$$ a\Big)\kern0.1em \left|k(y)\right|\ge 0 $$ is a piecewise continuous and bounded function in ℝ=−∞,∞$$ \mathbb{R}=\left(-\infty, \infty \right) $$. The coefficients by$$ b(y) $$ and qy$$ q(y) $$ are continuous functions in ℝ$$ \mathbb{R} $$ and can be unbounded at infinity. The operator L$$ L $$ admits closure in the space L2Ω$$ {L}_2\left(\Omega \right) $$, and the closure is also denoted by L$$ L $$. Taking into consideration certain constraints on the coefficients by$$ b(y) $$qy$$ q(y) $$, 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$$ s $$‐numbers). Here, we note that the estimate of singular numbers (s$$ s $$‐numbers) shows the rate of approximation of the resolvent of the operator L$$ L $$ by linear finite‐dimensional operators. It is given an example of how the obtained estimates for the s$$ s $$‐numbers enable to identify the estimates for the eigenvalues of the operator L$$ L $$. 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. [ABSTRACT FROM AUTHOR]