Threshold approximations for functions of a factorized operator family

In a Hilbert space $\mathfrak H$, we consider a family of selfadjoint operators (a quadratic operator pencil) $A(t)$, $t\in \mathbb{R}$, of the form $A(t) = X(t)^* X(t)$, where $X(t) = X_0 + t X_1$. It is assumed that the point $\lambda_0=0$ is an isolated eigenvalue of finite multiplicity for the operator $A(0)$. Let $F(t)$ be the spectral projection of the operator $A(t)$ for the interval $[0,\delta]$. Using approximations for $F(t)$ and $A(t)F(t)$ for $|t| \leqslant t_0$ (the so-called threshold approximations), we obtain approximations in the operator norm on $\mathfrak H$ for the operators $\cos ( \tau A(t)^{1/2})$ and $A(t)^{-1/2}\sin ( \tau A(t)^{1/2})$, $\tau \in \mathbb{R}$. The numbers $\delta$ and $t_0$ are controlled explicitly. Next, we study the behavior for small $\varepsilon >0$ of the operators $\cos (\varepsilon^{-1} \tau A(t)^{1/2})$ and $A(t)^{-1/2}\sin (\varepsilon^{-1}\tau A(t)^{1/2})$ multiplied by the “smoothing factor” $\varepsilon^q (t^2 + \varepsilon^2)^{-q/2}$ with a suitable $q>0$. The obtained approximations are given in terms of the spectral characteristics of the operator $A(t)$ near the lower edge of the spectrum. The results are aimed at application to homogenization of hyperbolic equations with periodic rapidly oscillating coefficients.