We consider vector periodic differential operator A admitting a factorization A=X*X, where X is a homogeneous differential operator of first order. Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral decomposition of A in a small neighborhood of zero are called threshold effects at the lower edge of the spectrum. An example of a threshold effect is the behavior of a differential operator in the small period limit. The ``effective characteristics'', namely, the homogenized medium, the effective mass, the effective 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.
Heinz Siedentop