In $L_2({ \mathbb R}^d;{ \mathbb C}^n)$, we consider a selfadjoint operator ${ \mathcal B}_\varepsilon$, $0< \varepsilon \leqslant 1$, given by the differential expression $b({ \mathbf D})^* g({ \mathbf x}/\varepsilon)b({ \mathbf D}) + \sum_{ j=1}^d (a_j({ \mathbf x}/\varepsilon) D_j +D_j a_j({ \mathbf x}/\varepsilon)^*) + Q({ \mathbf x}/\varepsilon)$, where $b({ \mathbf D}) = \sum_{ l=1}^d b_l D_l$ is the first order differential operator, and $g, a_j, Q$ are matrix-valued functions in ${ \mathbb R}^d$ periodic with respect to some lattice $\Gamma$. It is assumed that $g$ is bounded and positive definite, while $a_j$ and $Q$ are, in general, unbounded. We study the generalized resolvent $({ \mathcal B}_\varepsilon - \zeta Q_0({ \mathbf x}/\varepsilon))^{ -1}$, where $Q_0$ is a $\Gamma$-periodic, bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. Approximations for the generalized resolvent in the $(L_2 \to L_2)$- and $(L_2 \to H^1)$-norms with two-parametric error estimates (with respect to the parameters $\varepsilon$ and $\zeta$) are obtained.
Categories: math.AP