The time evolution of a class of meta-stable states

Abstract

The non-relativistic quantum mechanical description of meta-stable states which arise by perturbation of embedded eigenvalues is considered. The model given by the Hamiltonian

$$H = \left( {\begin{array}{*{20}c} { - \Delta } & {\lambda u} \\ {\lambda u} & { - \Delta + v} \\ \end{array} } \right)$$

is studied for small λ. If −Δ+v has a positive eigenvalue then when λ=0,H has an embedded eigenvalue. The corresponding eigenstate, Φ, is a meta-stable state for λє0. The time evolution of Φ underH,e ^−itH Φ is estimated uniformly int.