Second order periodic differential operators. Threshold properties and homogenization

The vector periodic differential operators (DO's) $\mathcal{A}$ admitting a factorization ${\mathcal{A}}={\mathcal{X}} ^*{\mathcal{X}}$, where $\mathcal{X}$ is a first order homogeneous DO, are considered in $L_2(\mathbb{R} ^d)$. Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral expansion of $\mathcal{A}$ 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 ${\mathcal{ A}}-\alpha V$, $\alpha >0$, where $V(\mathbf{x})\ge 0$ and $V(\mathbf{x}) \to 0$ as $\vert\mathbf{x}\vert\to \infty$. ``Effective characteristics'', such as the homogenized medium, effective mass, effective Hamiltonian, etc., arise in these problems. The general approach to these problems proposed in this paper is based on the spectral perturbation theory for operator-valued functions admitting analytic factorization. Most of the arguments are carried out in abstract terms. As to applications, the main attention is paid to homogenization of DO's.