# Abstract

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