Abstract
We determine the width of resonance-free domains in the complex plane for the semiclassical Schrödinger operator −h 2Δ+V(x) whenh→0, in terms of Lyapunov exponents for the associated classical flow.
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(D'autres références sur les résonances sont indiquées dans [5], et on se limite ici aux travaux qui jouent un rôle pour les techniques utilisées.)
Abraham, R., Marsden, J.: Foundations of mechanics. New York: Benjamin Cumming 1978
Abraham, R., Robbin, J.: Transversal mappings and flows. New York: Cummings 1967
Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J.21 (3), 193–226 (1971)
Gérard, C., Sjöstrand, J.: Semiclassical resonances generated by a closed trajectory of hyperbolic type. Commun. Math. Phys.108, 391–421 (1987)
Helffer, B., Sjöstrand, J.: Résonances en limite semiclassique. Bull. S.M.F., Mémoire Vol.24/25 (1986)
Hirsch, W.M., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Berlin, Heidelberg, New York: Springer 1977
Oseledets, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic exponents for dynamical systems. Trans. Moscow Math. Soc.19, 197–231 (1968)
Sjöstrand, J.: Singularités analytiques microlocales. Astérisque, Vol. 95 (1982)
Sjöstrand, J.: Semiclassical resonances generated by non-degenerate critical points. Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer 1987
Smale, S.: Differentiable dynamical systems. Bull. A.M.S., Vol. 73, 747–817 (1967)
Briet, Ph., Combes, J.M., Duclos, P.: On the location of resonances for Schrödinger operators in the semiclassical limit. II. Commun. PDE12 (2), 201–222 (1987)
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Communicated by B. Simon
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Gérard, C., Sjöstrand, J. Resonances en limite semiclassique et exposants de Lyapunov. Commun.Math. Phys. 116, 193–213 (1988). https://doi.org/10.1007/BF01225255
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DOI: https://doi.org/10.1007/BF01225255