Second microlocalisation

iε∂tψ^ε−|p^[|² 2 ψ^ε



= 1

|p^[|^1/2√

εR(θ0)^∗V0(q^[+ p^[(t − t^[)) R(θ0)u^ε+ O(ε^−1/2(t − t^[)²)

= 1

|p^[|^1/2√

εV0(|p^[|(t − t^[), ω ∧ q^[)u^ε+ O(√ εs²)

= V₀(s, η)u^ε+ O(√ εs²).

7.4.3 Second microlocalisation

The transition coefficient Tε brings light on this second scale√

ε. If we look at a functional of the form W^ε[ψ^ε(t)](x, ξ)T_ε(x, ξ)

and test it against a function a ∈ C₀^∞(R^2d, C^2,2), we see that it expresses as W^ε[ψ^ε(t)](x, ξ)T_ε(x, ξ) = (op_ε(a_ε)ψ^ε(t), ψ^ε(t)) with

aε(x, ξ) = a(x, ξ)Tε(x, ξ) = b



x, ξ,x ∧ ξ ε



since

Tε(x, ξ) = exp



−π ε

|x ∧ ξ|²

|ξ|³

 . The function

(x, ξ) 7→ x ∧ ξ

is a coordinate function in the open set R^d× (R^d\ {0}). It defines a hypersurface M = {x ∧ ξ = 0}

which has interesting properties.

Lemma 7.4.4. Any classical trajectory reaching the crossing set Υ is included in M . Proof. We observe that

d

dtx^±(t) ∧ ξ^±(t) = 0

for any classical trajectory. Therefore, if it passes at time t^[ through Υ, we have x^±(t) ∧ ξ^±(t) = x^±(t^[) ∧ ξ^±(t^[) = 0, ∀t ∈ R.

7.4. CODIMENSION 2 CROSSINGS 61 Therefore, the information concerning the concentration of ψ^ε(t) above the trajectories is contained in the information about the concentration of ψ^ε(t) above M . For analyzing this fact, one uses two-scale semi-classical pseudodifferential operators, two-scale Wigner transform and two-scale Wigner measures. It is for these two-scale Wigner measures associated with the concentration on M that one can write transition formula.

Let us first define two-scale observables. We extend the phase space T^∗R^d with a new variable η ∈ R, where R is the compactification of R^d obtained by adding two points at infinity, +∞ and −∞ The test functions associated with this extended phase space are functions a ∈ C^∞(T^∗R^dx,ξ× Rη) which satisfy the two following properties:

1. there exists a compact K ⊂ T^∗R^d such that, for all η ∈ R, the map (x, ξ) 7→ a(x, ξ, η) is a smooth function compactly supported in K;

2. there exists a function a_∞ defined on T^∗R^d∪ {+∞, −∞) and R0 > 0 such that, if |η| > R₀, then a(x, ξ, η) = a_∞(x, ξ, sgn(η)∞).

We denote by A the set of such functions.

We use the symbols of A to perform zooms along M at the scale √

ε. Then, for a ∈ A we write:

In addition, any function a ∈ C₀^∞(R^2d) can be naturally identified to a function in A which does not depend on the last variable. For such a, one has

W^M,ε[f ], a = (op_ε(a)f, f ) = hW^ε[f ], ai.

In terms of functionals, we can write

W^M,ε[f ], (x, ξ, η) = W^ε[f ] ⊗ δ One can then define two-scale Wigner measures.

Proposition 7.4.5. Let (f^ε)ε>0 be bounded in L²(R^d, C²); suppose in addition that this sequence has a semiclassical measure µ. Then, (W_f^M,εε )ε>0 is a bounded sequence in D⁰(R^d× R^d× R) whose accumulation points µ^M are matrix-valued positive measures on R^2d(x,ξ)× Rη that satisfy

µ(x, ξ) = Z

R

µ^M(x, ξ, dη).

62 CHAPTER 7. CONICAL INTERSECTIONS The measure µ^M decomposes into two parts: a compact part, which is essentially the restriction of µ^M to the interior of R^2dR which is the set R^2d× R, and a part at infinity, which corresponds to the restriction to the two points at infinity R^2d× {+∞, −∞}.

The transfer spells out nicely in terms of two-scale Wigner measures. We associate with (ψ^ε(t))_ε>0 the time-dependent two-scale Wigner measure µ^M(dx, dξ, dη, dt) associated with its concentration on M . As for the usual Wigner measure, we can prove that it decomposes on the modes: there exists two scalar positive measures µ^M,+and µ^M,−such that

µ^M(x, ξ, η, t) = µ^M,+(x, ξ, η, t)Π₊(x) + µ^M,−(x, ξ, η, t)Π₋(x).

Besides, outside Υ, µ^M,+and ν^M,− are absolutely continuous with respect to dt:

ν^M,±(x, ξ, η, t) = ν_t^M,±(x, ξ, η) ⊗ dt outside Υ,

and the measures µ^M,±_t propagates along classical trajectories outside Υ: assume Φ^t_±(x, ξ) /∈ Υ for all t ∈ [ti, tf] and (x, ξ) ∈ U , then for all a ∈ A compactly supported in U , we have

ha, µ^M,±_tf i = ha(Φ^ti−tf(x, ξ), η)µ^M,±_ti i.

In view of Lemma 7.4.1, the trajectories that pass through generic points of Υ comes from the region x · ξ < 0 to the region x · ξ > 0. Therefore, close to Υ, we set

µ^M,±_t,in = µ^M,−_t 1x·ξ<0, µ^M,±_t,out= µ^M,−_t 1x·ξ>0

for denoting the ingoing and outgoing two-scale Wigner measures. If µ^M,+_t,in and µ^M,−_t,in are singular, their trace on Σ = {x · ξ = 0} are given by the formula

µ^M,+_t,out(x, ξ, η) µ^M,−_t,out(x, ξ, η)

!

=



 1 − e^−π

η2

|ξ|3 e^−π

η2

|ξ|3

e^−π

η2

|ξ|3 1 − e^−π

η2

|ξ|3





µ^M,+_t,in (x, ξ, η) µ^M,−_t,in (x, ξ, η)

!

above Υ.

These types of results can be generalize in more general context. However, the set M is more complicated in the sense that there is a priori no simple equation for it. This is a specificity of the matrix V0 that we have chosen to consider.

Bibliography

[1] Serge Alinhac and Patrick G´erard. Pseudo-differential operators and the Nash-Moser theorem, volume 82 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2007. Translated from the 1991 French original by Stephen S. Wilson.

[2] Gr´egoire Allaire and Mariapia Palombaro. Localization for the Schr¨odinger equation in a locally periodic medium. SIAM J. Math. Anal., 38(1):127–142 (electronic), 2006.

[3] Gr´egoire Allaire and Andrey Piatnitski. Homogenization of the Schr¨odinger equation and effective mass theorems. Comm. Math. Phys., 258(1):1–22, 2005.

[4] Nalini Anantharaman, Fr´ed´eric Faure, and Clotilde Fermanian-Kammerer. Le chaos quantique. Actes des journ´ees X-UPS 2014, ´Editions de l’Ecole Polytechnique.

[5] Nalini Anantharaman, Clotilde Fermanian-Kammerer, and Fabricio Maci`a. Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures. Amer.

J. Math., 137(3):577–638, 2015.

[6] Nalini Anantharaman, Matthieu L´eautaud, and Fabricio Maci`a. Winger measures and observability for the Schr¨odinger equation on the disk. Invent. Math., 206(2):485–599, 2016.

[7] Nalini Anantharaman and Fabricio Maci`a. The dynamics of the Schr¨odinger flow from the point of view of semiclassical measures. In Spectral geometry, volume 84 of Proc. Sympos. Pure Math., pages 93–116.

Amer. Math. Soc., Providence, RI, 2012.

[8] Nalini Anantharaman and Fabricio Maci`a. Semiclassical measures for the Schr¨odinger equation on the torus. J. Eur. Math. Soc. (JEMS), 16(6):1253–1288, 2014.

[9] Nalini Anantharaman and Gabriel Rivi`ere. Dispersion and controllability for the Schr¨odinger equation on negatively curved manifolds. Anal. PDE, 5(2):313–338, 2012.

[10] Hajer Bahouri, Clotilde Fermanian Kammerer Isabelle Gallagher. Phase-space analysis and pseudod-ifferential calculus on the Heisenberg group, Ast´erisque, vol. 342, Soci´et´e Math´ematique de France, 2012.

[11] Luigi Barletti and Naoufel Ben Abdallah. Quantum transport in crystals: effective mass theorem and k·p Hamiltonians. Comm. Math. Phys., 307(3):567–607, 2011.

63

64 BIBLIOGRAPHY [12] Philippe Bechouche, Norbert J. Mauser, and Fr´ed´eric Poupaud. Semiclassical limit for the Schr¨

odinger-Poisson equation in a crystal. Comm. Pure Appl. Math., 54(7):851–890, 2001.

[13] Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou. Asymptotic analysis for periodic structures, volume 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York, 1978.

[14] Felix Bloch. ¨Uber die Quantenmechanik der Electronen in Kristallgittern. Z. Phys, 52:555–600, 1928.

[15] Max Born, Robert Oppenheimer. Zur Quantentheorie der Molekeln, Ann. der Phys. (4) 84 (1927), p. 457–484.

[16] Y. Colin de Verdi`ere: The level crossing problem in semi-classical analysis I. The symmetric case, Proceedings of Fr´ed´eric Pham’s congress, Annales de l’Institut Fourier (2002).

[17] Alberto-P. Calder´on and R´emi Vaillancourt. On the boundedness of pseudo-differential operators. J.

Math. Soc. Japan, 23:374–378, 1971.

[18] R´emi Carles and Christof Sparber. Semiclassical wave packet dynamics in Schr¨odinger equations with periodic potentials. Discrete Contin. Dyn. Syst. Ser. B, 17(3):759–774, 2012.

[19] Victor Chabu, Clotilde Fermanian-Kammerer, and Fabricio Maci`a. Semiclassical analysis of dispersion phenomena. 2017. to appear.

[20] Edward Brian Davies. Spectral theory and differential operators. Cambridge University Press, /newblock 1995.

[21] Mouez Dimassi, Jean-Claude. Guillot, and James Ralston. Gaussian beam construction for adiabatic perturbations. Math. Phys. Anal. Geom., 9(3):187–201 (2007), 2006.

[22] Mouez Dimassi and Johannes Sj¨ostrand. Spectral asymptotics in the semi-classical limit, volume 268 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1999.

[23] Thomas Duyckaerts, Thierry Jecko, Clotilde Fermanian Kammerer. Degenerated codimension 1 cross-ings and resolvent estimates, Asympt. Analysis, 3-4, p. 147-174 (2009) .

[24] Clotilde Fermanian-Kammerer. ´Equation de la chaleur et Mesures semi-classiques. PhD thesis, Univer-sit´e Paris-Sud, Orsay, 1995.

[25] Clotilde Fermanian-Kammerer. Mesures semi-classiques 2-microlocales. C. R. Acad. Sci. Paris S´er. I Math., 331(7):515–518, 2000.

[26] Clotilde Fermanian Kammerer: Analyse `a deux ´echelles d’une suite born´ee de L² sur une sous-vari´et´e du cotangent. C. R. Acad. Sci. Paris. 340, S´erie 1, p. 269-274 (2005).

[27] C. Fermanian Kammerer and P. G´erard: Mesures semi-classiques et croisements de modes, Bull. Soc.

Math. Fr., 130, no. 1 (2002), pp. 123–168.

[28] C. Fermanian Kammerer and P. G´erard: A Landau-Zener formula for non-degenerated involutive codi-mension 3 crossings, Ann. Henri Poincar´e, 4, no. 3, pp. 513–552 (2003).

BIBLIOGRAPHY 65 [29] Clotilde Fermanian Kammerer, Caroline Lasser and Didier Robert, Adiabatic and non-adiabatic

evolu-tion of wave packets and applicaevolu-tions to initial value representaevolu-tions. arXiv:2011.01618

[30] Clotilde Fermanian Kammerer and Caroline Lasser, Propagation through generic level crossings: a surface hopping semigroup, SIAM J. of Math. Anal., 140, 1, p. 103-133 (2008).

[31] Clotilde Fermanian Kammerer and Caroline Lasser, Single switch surface hopping for molecular dynam-ics, J Math. Chem., 50, 620–635 (2012).

[32] Domenico Fiorenza, Domenico Monaco and Gianluca Panati. Z² invariants of topological insulators as geometric obstructions. Commun. Math. Phys. 343, 1115–1157 (2016).

[33] V´eronique Fischer, Michael Ruzhansky. Quantization on nilpotent Lie groups, Progress in Mathematics, 314, Birkh¨auser Basel, 2016.

[34] Gaston Floquet. Sur les ´equations diff´erentielles lin´eaires `a coefficients p´eriodiques. Ann. Sci. ´Ecole Norm. Sup. (2), 12:47–88, 1883.

[35] Gerald B. Folland. Harmonic analysis in phase space, volume 122 of Annals of Mathematics Studies.

Princeton University Press, Princeton, NJ, 1989.

[36] Christian G´erard, Andr´e Martinez, and Johannes Sj¨ostrand. A mathematical approach to the effective Hamiltonian in perturbed periodic problems. Comm. Math. Phys., 142(2):217–244, 1991.

[37] Patrick G´erard. Mesures semi-classiques et ondes de Bloch. In S´eminaire sur les ´Equations aux D´eriv´ees Partielles, 1990–1991, pages Exp. No. XVI, 19. ´Ecole Polytech., Palaiseau, 1991.

[38] Patrick G´erard. Microlocal defect measures. Comm. Partial Differential Equations, 16(11):1761–1794, 1991.

[39] Patrick G´erard and ´Eric Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem.

Duke Math. J., 71(2):559–607, 1993.

[40] Patrick G´erard, Peter A. Markowich, Norbert J. Mauser, and Fr´ed´eric Poupaud. Homogenization limits and Wigner transforms. Comm. Pure Appl. Math., 50(4):323–379, 1997.

[41] George A. Hagedorn: Molecular Propagation through Electron Energy Level Crossings. Memoirs of the A. M. S., 111, N^o 536, (1994).

[42] G. A. Hagedorn, A. Joye: Landau-Zener transitions through small electronic eigenvalue gaps in the Born-Oppenheimer approximation. Ann. Inst. Henri Poincar´e, 68, N^o1, p. 85-134 (1998).

[43] G. A. Hagedorn, A. Joye: Molecular propagation through small avoided crossings of electron energy levels. Rev. Math. Phys., 1, N^o1, p. 41-101 (1999).

[44] Bernard Helffer. 30 ans d’analyse semi-classique, bibliographie comment´ee, http://www.math.u-psud.

fr/~helffer/histoire2003.ps.

[45] Bernard Helffer and Johannes Sj¨ostrand. Equation de Schr¨odinger avec champ magn´etique et ´equation de Harper, volume 345 of Lecture Notes in Physics. Springer Verlag, 1989.

66 BIBLIOGRAPHY [46] Morris W. Hirsch. Differential topology, volume 33 of Graduate Texts in Mathematics. Springer-Verlag,

New York, 1994. Corrected reprint of the 1976 original.

[47] Mark A. Hoefer and Michael I. Weinstein. Defect modes and homogenization of periodic Schr¨odinger operators. SIAM J. Math. Anal., 43(2):971–996, 2011.

[48] Lars H¨ormander. The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979), no. 3, 360–444.

[49] Frank H¨overmann, Herbert Spohn, and Stefan Teufel. Semiclassical limit for the Schr¨odinger equation with a short scale periodic potential. Comm. Math. Phys., 215(3):609–629, 2001.

[50] I.L. Hwang : the L²boundedness of pseudo-differential operators. Trans. Amer. Math. Soc. 302, (1987), p.55-76.

[51] Thierry Jecko. Estimations de la r´esolvante pour une mol´ecule diatomique dans l’approximation de Born-Oppenheimer. Comm. Math. Phys. 195 (1998) no. 3, p. 585–612.

[52] Thierry Jecko. Semiclassical resolvent estimates for Schr¨odinger matrix operators with eigenvalues cross-ings. Math. Nachr., 257 (2003), no. 1, p. 36-54.

[53] Thierry Jecko. From classical to semiclassical non-trapping behaviour. C. R. Acad. Sci. Paris, Ser. I 338 (2004), p. 545–548.

[54] Thierry Jecko. Non-trapping condition for semiclassical Schr¨odinger operators with matrix-valued po-tentials. Math. Phys. Electronic Journal 11 (2005), no. 2. Erratum: Math. Phys. Electronic Journal, No.

3, vol. 13, 2007.

[55] Peter Kuchment. The mathematics of photonic crystals. In Mathematical modeling in optical science, volume 22 of Frontiers Appl. Math., pages 207–272. SIAM, Philadelphia, PA, 2001.

[56] Peter Kuchment. On some spectral problems of mathematical physics. In Partial differential equations and inverse problems, volume 362 of Contemp. Math., pages 241–276. Amer. Math. Soc., Providence, RI, 2004.

[57] Peter Kuchment. An overview of periodic elliptic operators. Bull. Amer. Math. Soc. (N.S.), 2016. To appear.

[58] L. Landau: Collected papers of L. Landau, Pergamon Press, (1965).

[59] C. Lasser and W. Gaim: Corrections to Wigner type phase space methods Nonlinearity 27, pp. 2951-2974 (2014).

[60] C. Lasser and J. Keller. Propagation of Quantum Expectations with Husimi Functions SIAM J. Appl.

Math. 73, pp. 1557-1581 (2013).

[61] C. Lasser, J. Keller and T. Ohsawa: A new phase space density for quantum expectations SIAM J.

Math. An. 48(1), pp. 513–537 (2016)

BIBLIOGRAPHY 67 [62] C. Lasser and S. Kube, M. Weber. Monte Carlo sampling of Wigner functions and surface hopping

quantum dynamics J. Comput. Phys. 228, pp. 1947-1962 (2009).

[63] L. Landau: Collected papers of L. Landau, Pergamon Press, (1965).

[64] Nicolas Lerner. Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications, vol. 3, Birkh¨auser Verlag, Basel, 2010.

[65] Mathieu Lewin. ´El´ements de th´eorie spectrale: le Laplacien sur un ouvert born´e. Master. France. 2017.

¡cel-01490197v2¿

[66] Pierre-Louis Lions and Thierry Paul. Sur les mesures de Wigner. Rev. Mat. Iberoamericana, 9(3):553–

618, 1993.

[67] Joachim Mazdak Luttinger and Walter Kohn. Motion of electrons and holes in perturbed periodic fields.

Phys. Rev., 97:869–883, Feb 1955.

[68] Fabricio Maci`a. Semiclassical measures and the Schr¨odinger flow on Riemannian manifolds. Nonlinearity, 22(5):1003–1020, 2009.

[69] Fabricio Maci`a. High-frequency propagation for the Schr¨odinger equation on the torus. J. Funct. Anal., 258(3):933–955, 2010.

[70] Fabricio Maci`a. The Schr¨odinger flow on a compact manifold: High-frequency dynamics and dispersion.

In Modern Aspects of the Theory of Partial Differential Equations, volume 216 of Oper. Theory Adv.

Appl., pages 275–289. Springer, Basel, 2011.

[71] Fabricio Maci`a. High-frequency dynamics for the Schr¨odinger equation, with applications to dispersion and observability. In Nonlinear optical and atomic systems, volume 2146 of Lecture Notes in Math., pages 275–335. Springer, Cham, 2015.

[72] Fabricio Maci`a and Gabriel Rivi`ere. Concentration and non-concentration for the Schr¨odinger evolution on Zoll manifolds. Comm. Math. Phys., 345(3):1019–1054, 2016.

[73] Andr´e Martinez. An introduction to semiclassical and microlocal analysis. Universitext. Springer-Verlag, New York, 2002.

[74] Abderrahim Outassourt. Comportement semi-classique pour l’op´erateur de Schr¨odinger `a potentiel p´eriodique. J. Funct. Anal., 72(1):65–93, 1987.

[75] Gianluca Panati, Herbert Spohn, and Stefan Teufel. Effective dynamics for Bloch electrons: Peierls substitution and beyond. Comm. Math. Phys., 242(3):547–578, 2003.

[76] Fr´ed´eric Poupaud and Christian Ringhofer. Semi-classical limits in a crystal with exterior potentials and effective mass theorems. 1996.

[77] Michael Reed and Barry Simon. Methods of modern mathematical physics. I to IV. Analysis of operators.

Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.

68 BIBLIOGRAPHY [78] Michael Ruzhansky, Ville Turunen. On the Fourier analysis of operators on the torus, in Modern trends in pseudo-differential operators, Oper. Theory Adv. Appl., vol. 172, Birkh¨auser, Basel, 2007, 87–105.

[79] Mikhail A. Shubin. Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathe-matics, Springer-Verlag, Berlin, 1987, in russian: 1978.

[80] Andr´e Voros. D´eveloppements semi-classiques, th`ese d’´etat, 1977.

[81] Alexander Watson, and Michael I. Weinstein. Wavepackets in Inhomogeneous Periodic Media: Propa-gation Through a One-Dimensional Band Crossing. Comm. Math. Phys. 363(2): 655–698, 2018.

[82] Eugene P. Wigner : Group Theory and its Application to the Quantum Mechanics of Atomic Spectra.

New York, Academic Press, 1959.

[83] Calvin H. Wilcox. Theory of Bloch waves. J. Analyse Math., 33:146–167, 1978.

[84] C. Zener: Non-adiabatic crossing of energy levels, Proc. Roy. Soc. Lond. 137, (1932), pp. 696-702.

[85] Maciej Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.

In document Multi-scale analysis and Schrödinger operators (Page 60-68)