Threshold approximations for a factorized selfadjoint operator family with the first and second correctors taken into account

Abstract:

In a Hilbert space $\mathfrak {H}$, a family of operators $A(t)$ admitting a factorization of the form $A(t)= X(t)^*X(t)$, where $X(t)=X_0 +tX_1$, $t \in \mathbb {R}$, is considered. It is assumed that the point $\lambda _0=0$ is an isolated eigenvalue of finite multiplicity for $A(0)$. Let $F(t)$ be the spectral projection of $A(t)$ for the interval $[0,\delta ]$ (where $\delta$ is sufficiently small). For small $|t|$, approximations in the operator norm in $\mathfrak {H}$ are obtained for the projection $F(t)$ with an error of $O(|t|^3)$ and for the operator $A(t)F(t)$ with an error of $O(|t|^5)$ (the threshold approximations). By using these results, approximation in the operator norm in $\mathfrak {H}$ are constructed for the operator exponential $\exp (-A(t)\tau )$ for large $\tau >0$ with an error of $O(\tau ^{-3/2})$. For the resolvent $(A(t)+\varepsilon ^2 I)^{-1}$ multiplied by a suitable “smoothing” factor, approximation in the operator norm in $\mathfrak {H}$ for small $\varepsilon >0$ with an error of $O(\varepsilon )$ is obtained. All approximations are given in terms of the spectral characteristics of $A(t)$ near the bottom of the spectrum. In these approximations, the first and the second correctors are taken into account. The results are aimed at applications to homogenization problems for periodic differential operators in the small period limit.