Threshold approximations with corrector for the resolvent of a factorized selfadjoint operator family

Abstract:

In a Hilbert space, a family of operators admitting a factorization $A(t)= X(t)^*X(t)$, where $X(t)=X_0 +tX_1$, $t \in \mathbb R$, is considered. It is assumed that the subspace $\mathfrak {N} = \operatorname {Ker} A(0)$ is finite-dimensional. For the resolvent $(A(t)+\varepsilon ^2 I)^{-1}$ with small $\varepsilon$, an approximation in the operator norm is obtained on a fixed interval $|t| \le t^0$. This approximation involves the so-called “corrector”; the remainder term is of order $O(1)$. The results are aimed at applications to homogenization of periodic differential operators in the small period limit. The paper develops and refines the results of Chapter 1 of our paper in St. Petersburg Math. J. 15 (2004), 639–714.