| Abstrakt: |
LetH(x,D, ?) be a self-adjoint partial differential operator of the form $$H = \sum\limits_{k = 0}^K {\varepsilon ^k H_k (x,\varepsilon D),{\rm{ }}x \in R^n }$$ . Suppose the hamiltonian system $$\dot x = \frac{{\partial H_0 }}{{\partial \xi }},{\rm{ }}\dot \xi = - \frac{{\partial H_0 }}{{\partial x}}$$ has a nondegenerate stable periodic orbit ? on which Then it is possible to construct a sequence of real numbers ?m tending to zero, a sequence of functionsum concentrated in a tube of radius ?m1/2 about the projection of ? intox-space, and a polynomialE(?) such that $$\parallel (H(\varepsilon _m ) - E(\varepsilon _m ))u_m \parallel \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} C\varepsilon _m^M \parallel u_m \parallel$$ . The powerM depends on the order of stability of ?. The constructions are explicit in terms of solutions of linear O.D.E.'s, and are generalizations of “gaussian beams”. Actually, instead of just one sequence, one gets a family of sequences parametrized by the multi-indices of ordern-1, but the constantC is not independent of these multi-indices. The nondegeneracy hypothesis implies ? is part of a one-parameter family of stable periodic orbits, andC is independent of this parameter. |
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