Approximation of the resolvent of a two-parametric quadratic operator pencil near the bottom of the spectrum

Abstract:

A two-parametric pencil of selfadjoint operators $B(t,\varepsilon )=X(t)^* X(t) + \varepsilon (Y_2^* Y(t)+Y(t)^*Y_2) + \varepsilon ^2 Q$ in a Hilbert space is considered, where $X(t)=X_0+tX_1$, $Y(t)=Y_0+tY_1$. It is assumed that the point $\lambda _0=0$ is an isolated eigenvalue of finite multiplicity for the operator $X_0^*X_0$, and that the operators $Y(t)$, $Y_2$, and $Q$ are subordinate to $X(t)$ in a certain sense. The object of study is the generalized resolvent $(B(t,\varepsilon )+\lambda \varepsilon ^2 Q_0)^{-1}$, where the operator $Q_0$ is bounded and positive definite. Approximation of this resolvent is obtained for small $\tau = (t^2 + \varepsilon ^2)^{1/2}$ with an error term of $O(1)$. This approximation is given in terms of some finite rank operators and is the sum of the principal term and the corrector. The results are aimed at applications to homogenization problems for periodic differential operators in the small period limit.