M.Sh. Birman, T.A. Suslina
Second Order Periodic Differential Operators. Threshold Properties and Homogenization
Preprint series: ESI preprints

MSC:

35B27 Homogenization; partial differential equations in media with periodic structure, See also {73B27, 76D30}

Abstract: In $L_2(\R^d)$, we consider vector periodic differential operators (DO's) $\cal A$ admitting a factorization ${\cal A}={\cal X}^*{\cal X}$, where $\cal X$ is a first order homogeneous DO. Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral decomposition of $\cal A$ in a small neighborhood of zero are called threshold effects at the point $\lambda=0$. An example of a threshold effect is the behavior of a DO in the small period limit (the homogenization effect). Another example is related to the negative discrete spectrum of the operator ${\cal A}-\alpha V$, $\alpha >0$, where $V(\x)\ge 0$ and $V(\x)\to 0$ as $|\x|\to \infty$. The
"effective characteristics", namely, the homogenized medium, the effective mass and Hamiltonian, etc., arise in these problems.
We propose a general approach to these problems based on the spectral perturbation theory for operator-valued functions admitting analytic factorization. A great deal of considerations is fulfilled in abstract terms. As for applications, the main attention is paid to the homogenization of DO's.
Keywords: Periodic operators, threshold effect, homogenization