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Abstract

In this article we study the propagation of Wigner measures linked to solutions of the Schrödinger equation with potentials presenting conical singularities and show that they are transported by two different Hamiltonian flows, one over the bundle cotangent to the singular set and the other elsewhere in the phase space, up to a transference phenomenon between these two regimes that may arise whenever trajectories in the outsider flow lead in or out the bundle. We describe in detail either the flow and the mass concentration around and on the singular set and illustrate with examples some issues raised by the lack of unicity for the classical trajectories at the singularities despite the unicity of the quantum solutions, dismissing any classical selection principle, but in some cases being able to fully solve the propagation problem.

Keywords

Schrödinger equation Wigner measures two-microlocal measures symbolic calculus quantum-classical correspondence wave packet approximation

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References

Ambrosio and

Figalli, Almost everywhere well-posedness of continuity equations with measure initial data, C. R. Math. Acad. Sci. Paris 1(348) (2010), 249–252. doi:10.1016/j.crma.2010.01.018.

Ambrosio,

Figalli,

Friesecke,

Giannoulis and

Paul, Semiclassical limit of quantum dynamics with rough potentials and well-posedness of transport equations with measure initial data, Comm. Pure. Appl. Math. 64 (2011), 1199–1242. doi:10.1002/cpa.20371.

Athanassoulis, Generalized wave packets for the semiclassical Schrödinger equation with power non-linearities, 2015 (arXiv:1505.04707 [math.AP]).

Athanassoulis and

Paul, Strong and weak semiclassical limits for some rough Hamiltonians, Math. Models Methods Appl. Sci. 22 (2012), 1250038. doi:10.1142/S0218202512500388.

Athanassoulis and

Paul, On the selection of the classical limit for potentials with BV derivatives, J. Dyn. Diff. Equat. 25 (2013), 33–47. doi:10.1007/s10884-012-9284-z.

Chabu, Semiclassical analysis of the Schrödinger equation with irregular potentials, Créteil, 2016, Thèse de l’Université Paris-Est.

Combescure and

Robert, Coherent States and Applications in Mathematical Physics, Springer Science & Business, Media, 2012.

Dimassi and

Sjöstrand, Spectral Asymptotics in the Semiclassical Limit, Cambridge University Press, Cambridge 1999.

Fermanian, Mesures semiclassiques 2-microlocales, C. R. Acad. Sci. Paris 1(331) (2000), 515–518. doi:10.1016/S0764-4442(00)01660-8.

10.

Fermanian, Opérateurs pseudodifférentiels semiclassiques, in: Chaos en Mécanique Quantique,

Harinck,

Plagne and

Sabbah, eds, École Polytechnique, Palaiseau, 2014.

11.

Fermanian, Two-scale analysis of a bounded family in

L^{2}

on a submanifold of the phase space, C. R. Acad. Sci. Paris 1(340) (2005), 269–274. doi:10.1016/j.crma.2005.01.002.

12.

Fermanian,

Gérard and

Lasser, Wigner measure propagation and conical singularity for general initial data, Arch. Ration. Mech. Anal. 209 (2012), 209–236. doi:10.1007/s00205-013-0622-z.

13.

Fermanian and

Lasser, Wigner measures and codimension two crossings, J. Math. Phys. 44 (2003), 507–527. doi:10.1063/1.1527221.

14.

Figalli,

Ligabò and

Paul, Semiclassical limit for mixed states with singular and rough potentials, Indiana Univ. Math. Journal 61 (2012), 193–222. doi:10.1512/iumj.2012.61.4670.

15.

Gérard, Mesures Semiclassiques et Ondes de Bloch, Séminaire sur les Équations aux Dérivées Partielles 1990-91, XVI, École Polytechnique, Palaiseau, 1991.

16.

Gérard,

Markowich,

Mauser and

Poupaud, Homogeneization limits and Wigner transforms, Comm. Pure. Appl. Math. 50 (1997), 323–379, Comm. Pure. Appl. Math.53, 280–281, 2000. doi:10.1002/(SICI)1097-0312(200002)53:2<280::AID-CPA4>3.0.CO;2-6.

17.

Hagedorn, Semiclassical quantum mechanics, Comm. Math. Phys. 71 (1980), 77–93. doi:10.1007/BF01230088.

18.

Kato, Perturbation Theory for Linear Operators, Springer Verlag, Berlin Heidelberg, 1995.

19.

Lions and

Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9 (1993), 553–618. doi:10.4171/RMI/143.

20.

Miller, Propagation d’ondes semiclassiques à travers une interface et mesures 2-microlocales, Thèse de l’École Polytechnique, Palaiseau, 1996.

21.

Nier, A semiclassical picture of quantum scatering, Ann. Sci. Ecole Norm. Sup. 4 29 (1996), 149–183.

22.

Reed and

Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, San Diego 1975.

23.

Schubert,

Vallejos and

Toscano, How do wave packets spread? Time evolution on Ehrenfest time scales, J. Phys. A 45 (2012), 215307. doi:10.1088/1751-8113/45/21/215307.

24.

Tartar, H-measures, a new approach for studying homogenization, oscillation and concentration effects in partial differential equations, Proc. Roy. Soc. Edinb. A 115 (1990), 193–230. doi:10.1017/S0308210500020606.

25.

Zworski, Semiclassical Analysis, American Mathematical Society, Providence, 2012.