A general effective Hamiltonian method

Abstract

We perform a general reduction scheme that can be applied in particular to the spectral study of operators of the type $P=P(x,y,h{D}_x,D_y)$ as $h$ tends to zero. This scheme permits to reduce the study of $P$ to the one of a semiclassical matrix operator of the type $A=A(x,h{D}_x)$. Here, for any fixed $(x,\xi )\in {\mathbb{R}}^n$, the eigenvalues of the principal symbol $a(x,\xi )$ of $A$ are eigenvalues of the operator $P(x,y,\xi ,D_y)$.