Threshold Effects near the Lower Edge of the Spectrum for Periodic Differential Operators of Mathematical Physics

Abstract

In L ₂ $\left( {{\mathbb{R}^d}} \right),$ we consider vector periodic DO A admitting a factorization A = X*X, where X is a homogeneous DO 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 λ = O. An example of a threshold effect is the behavior of a DO in the small period limit Another example is related to the negative discrete spectrum of the operator A- α V, α> 0, where V(x) ≥ 0 and V(x) → 0 as |x|→ ∞. 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. A great deal of considerations is done in abstract terms.