Two-scale Wigner Measures and the Landau—Zener Formulas

Abstract

We consider the system of evolution equations

$$i\varepsilon \frac{{\partial {{\psi }^{\varepsilon }}}}{{{{\partial }_{t}}}} = o{{p}_{\varepsilon }}(H){{\psi }^{\varepsilon }},$$

where ε is a small parameter, ψ^ε=ψ^ε(t,x) a vector-valued bounded family in L ²(ℝ^d), H = H(t,x,ξ) a matrix-valied Hamiltonian.The variable x denotes the position variable and ξ the momentum.We use Weyl quantization:

$$o{{p}_{\varepsilon }}(a) = \int_{{{{\mathbb{R}}^{d}} \times {{\mathbb{R}}^{d}}}} {{{e}^{{i(x - y) \cdot \xi }}}a} \left( {\frac{{x + y}}{2},\varepsilon \xi } \right)f(y)\frac{{dyd\xi }}{{{{{(2\pi )}}^{d}}}}.$$