Approximations of the Resolvent for a Non–Self-Adjoint Diffusion Operator with Rapidly Oscillating Coefficients

A strongly inhomogeneous diffusion operator with drift depending on a small parameter $\varepsilon$ is studied in the space $L^2(\mathbb R^n)$. The strong inhomogeneity consists in that the coefficients of the operator are $\varepsilon$-periodic and, in addition, the drift vector is of the order of $\varepsilon^{-1}$. As $\varepsilon\to 0$, approximations in the operator $L^2$‑norm of order $\varepsilon$ and $\varepsilon^2$ are constructed for the resolvent of the operator. For each of these orders of approximation, an averaged diffusion operator is obtained. A spectral method based on the Bloch representation for an operator with periodic coefficients is used.