Two scale analysis of a bounded family in \(L^2\) on a submanifold of the phase space. (Analyse à deux échelles d’une suite bornée de \(L^2\) sur une sous-variété du cotangent.) (French. Abridged English version) Zbl 1067.35083
Summary: We investigate the generalization of two-scale Wigner measure to the case of submanifolds more general than symplectic and involutive ones for which they have been defined. We study the concentration of a bounded family in \(L^2(\mathbb R^d)\) on a submanifold of the cotangent space \(T^{\ast}\mathbb R^d\) for which the restriction of the symplectic form to its tangent space is of constant rank.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 53D50 | Geometric quantization |
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
Keywords:
symplectic submanifolds; Schrödinger equation; Wigner measure; cotangent space; symplectic formReferences:
| [1] | Fermanian Kammerer, C., Mesures semi-classiques 2-microlocales, C. R. Acad. Sci. Paris, Ser. I, 331, 515-518 (2000) · Zbl 0964.35009 |
| [2] | Fermanian Kammerer, C., A non commutative Landau-Zener formula, Math. Nach., 271, 22-50 (2004) · Zbl 1050.35093 |
| [3] | Fermanian Kammerer, C., Semi-classical analysis of generic codimension 3 crossings, Int. Math. Res. Not., 45, 2391-2435 (2004) · Zbl 1098.81038 |
| [4] | Fermanian Kammerer, C.; Gérard, P., Mesures semi-classiques et croisements de modes, Bull. Soc. Math. France, 130, 1, 145-190 (2002) |
| [5] | Fermanian Kammerer, C.; Gérard, P., A Landau-Zener formula for non-degenerated involutive codimension 3 crossings, Ann. Inst. H. Poincaré, 4, 513-552 (2003) · Zbl 1049.81029 |
| [6] | P. Gérard, Mesures semi-classiques et ondes de Bloch, Exposé de l’Ecole Polytechnique, E.D.P., Exposé N XVI, 1991; P. Gérard, Mesures semi-classiques et ondes de Bloch, Exposé de l’Ecole Polytechnique, E.D.P., Exposé N XVI, 1991 |
| [7] | Gérard, P.; Leichtnam, E., Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., 71, 559-607 (1993) · Zbl 0788.35103 |
| [8] | Gérard, P.; Markowich, P. A.; Mauser, N. J.; Poupaud, F., Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50, 4, 323-379 (1997) · Zbl 0881.35099 |
| [9] | Helffer, B.; Martinez, A.; Robert, D., Ergodicité et limite semi-classique, Comm. Math. Phys., 109, 313-326 (1987) · Zbl 0624.58039 |
| [10] | Hörmander, L., The Analysis of Linear Partial Differential Operators III (1985), Springer-Verlag · Zbl 0601.35001 |
| [11] | Lions, P.-L.; Paul, T., Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9, 553-618 (1993) · Zbl 0801.35117 |
| [12] | L. Miller, Propagation d’onde semi-classiques à travers une interface et mesures 2-microlocales, Thèse de l’Ecole Polytechnique, 1996; L. Miller, Propagation d’onde semi-classiques à travers une interface et mesures 2-microlocales, Thèse de l’Ecole Polytechnique, 1996 |
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