Second order periodic differential operators. Threshold properties and homogenization
M. Sh. Birman, Tatiana Aleksandrovna Suslina · St Petersburg Mathematical Journal · 2004
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