Semi-classical analysis on H-type groups Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday
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摘要
In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.
In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.
In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.
引文
1 Anantharaman N,Faure F,Fermanian Kammerer C.Chaos Quantique.Palaiseau:′Ecole Polytechnique,2014
2 Anantharaman N,Maci`a F.The dynamics of the Schr¨odinger flow from the point of view of semiclassical measures.In:Spectral Geometry.Proceedings of Symposia in Pure Mathematics,vol.84.Providence:Amer Math Soc,2012,93-116
3 Anantharaman N,Maci`a F.Semiclassical measures for the Schr¨odinger equation on the torus.J Eur Math Soc(JEMS),2014,16:1253-1288
4 Astengo F,Di Blasio B,Ricci F.Gelfand pairs on the Heisenberg group and Schwartz functions.J Funct Anal,2009,256:1565-1587
5 Bahouri H,Chemin J-Y,Danchin R.A frequency space for the Heisenberg group.ArXiv:1609.03850,2016
6 Bahouri H,Chemin J-Y,Danchin R.Tempered distributions and Fourier transform on the Heisenberg group.ArX-iv:1705.02195,2017
7 Bahouri H,Chemin J-Y,Gallagher I.Precised Hardy inequalities on Rdand on the Heisenberg group Hd.S′eminaire′Equations aux D′eriv′ees Partielles de l’′Ecole Polytechnique(2004-2005),Expos′e XIX,2005
8 Bahouri H,Chemin J-Y,Xu C-J.Trace theorem in Sobolev spaces associated with H¨ormander’s vector fields.In:Partial Differential Equations and Their Applications.River Edge:World Sci Publ,1999,1-14
9 Bahouri H,Chemin J-Y,Xu C-J.Trace theorem on the Heisenberg group on homogeneous hypersurfaces.In:Phase Space Analysis of Partial Differential Equations.Progress in Nonlinear Differential Equations and Their Applications,vol.69.Boston:Birkh¨auser,2006,1-15
10 Bahouri H,Chemin J-Y,Xu C-J.Trace theorem on the Heisenberg group.Ann Inst Fourier(Grenoble),2009,59:491-514
11 Bahouri H,Fermanian Kammerer C,Gallagher I.Phase Space Analysis on the Heisenberg Group.Ast′erisque,vol.345.Paris:Soc Math France,2012
12 Bahouri H,Fermanian Kammerer C,Gallagher I.Dispersive estimates for the Schr¨odinger operator on step 2 stratified Lie groups.Anal PDE,2016,9:545-574
13 Bahouri H,G′erard P,Xu C-J.Espaces de Besov et estimations de Strichartz g′en′eralis′ees sur le groupe de Heisenberg.J Anal Math,2000,82:93-118
14 Beals R,Greiner P.Calculus on Heisenberg Manifolds.Annals of Mathematics Studies,vol.119.Princeton:Princeton University Press,1988
15 Christ M,Geller D,Glowacki P,et al.Pseudodifferential operators on groups with dilations.Duke Math J,1992,68:31-65
16 Colin de Verdi`ere Y,Hillairet L,Tr′elat E.Spectral asymptotics for sub-Riemannian Laplacians.I:Quantum ergodicity and quantum limits in the 3D contact case.Duke Math J,2018,167:109-174
17 Damek E,Ricci F.Harmonic analysis on solvable extensions of H-type groups.J Geom Anal,1992,2:213-248
18 Del Hierro M.Dispersive and Strichartz estimates on H-type groups.Studia Math,2005,169:1-20
19 Dixmier J.C*-Algebras.North-Holland Mathematical Library,vol.15.Amsterdam-New York-Oxford:North-Holland,1977
20 Faraut J,Harzallah K.Deux Cours d’Analyse Harmonique.Progress in Mathematics,vol.69.Boston:Birkh¨auser,1987
21 Fermanian Kammerer C,Fischer V.Defect measures on graded Lie groups.ArXiv:1707.04002,2017
22 Fermanian Kammerer C,G′erard P.A Landau-Zener formula for non-degenerated involutive codimension 3 crossings.Ann Henri Poincar′e,2003,4:513-552
23 Fischer V,Ricci F,Yakimova O.Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I:Rank-one actions on the centre.Math Z,2012,271:221-255
24 Fischer V,Ruzhansky M.Quantization on Nilpotent Lie Groups.Progress in Mathematics,vol.314.Basel:Birkh¨auser,2016
25 Folland G B,Stein E.Hardy Spaces on Homogeneous Groups.Princeton:Princeton University Press,1982
26 Geller D.Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability.Mathematical Notes,vol.37.Princeton:Princeton University Press,1990
27 G′erard P.Mesures semi-classiques et ondes de Bloch.S′eminaire′Equations aux D′eriv′ees Partielles de l’′Ecole Polytechnique(1990-1991),Expos′e XVI,1991
28 G′erard P,Leichtnam E.Ergodic properties of eigenfunctions for the Dirichlet problem.Duke Math J,1993,71:559-607
29 G′erard P,Markowich P A,Mauser N J,et al.Homogenization limits and Wigner transforms.Comm Pure Appl Math,1997,50:323-379
30 Helffer B.30 ans d’analyse semi-classique:Bibliographie comment′ee.Http://www.math.u-psud.fr/～helffer/histoire2003.ps
31 Helffer B J,Sj¨ostrand J.Equation de Schr¨odinger avec champ magn′etique et′equation de Harper.In:Lecture Notes in Physics,vol.345.Berlin:Springer,1989,118-198
32 H¨ormander L.The Weyl calculus of pseudodifferential operators.Comm Pure Appl Math,1979,32:360-444
33 Kaplan A.Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms.Trans Amer Math Soc,1980,258:147-153
34 Kor′anyi A.Some applications of Gelfand pairs in classical analysis.In:Harmonic Analysis and Group Representations.Naples:Liguori,1982,333-348
35 L′evy G.The Fourier transform on 2-step Lie groups.ArXiv:1712.09880,2017
36 Lions P-L,Paul T.Sur les mesures de Wigner.Rev Mat Iberoamericana,1993,9:553-618
37 Mantoiu M,Ruzhansky M.Pseudo-differential operators,Wigner transform and Weyl systems on type I locally compact groups.Doc Math,2017,22:1539-1592
38 Ruzhansky M,Turunen V.On the Fourier analysis of operators on the torus.In:Modern Trends in Pseudo-Differential Operators.Operator Theory:Advances and Applications,vol.172.Basel:Birkh¨auser,2007,87-105
39 Shubin M L A.Pseudodifferential Operators and Spectral Theory.Springer Series in Soviet Mathematics.Berlin:Springer-Verlag,1987
40 Taylor M E.Noncommutative Microlocal Analysis.I.Memoirs of the American Mathematical Society,vol.52.Providence:Amer Math Soc,1984
41 Wigner E O.Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra.New York:Academic Press,1959
2 Anantharaman N,Maci`a F.The dynamics of the Schr¨odinger flow from the point of view of semiclassical measures.In:Spectral Geometry.Proceedings of Symposia in Pure Mathematics,vol.84.Providence:Amer Math Soc,2012,93-116
3 Anantharaman N,Maci`a F.Semiclassical measures for the Schr¨odinger equation on the torus.J Eur Math Soc(JEMS),2014,16:1253-1288
4 Astengo F,Di Blasio B,Ricci F.Gelfand pairs on the Heisenberg group and Schwartz functions.J Funct Anal,2009,256:1565-1587
5 Bahouri H,Chemin J-Y,Danchin R.A frequency space for the Heisenberg group.ArXiv:1609.03850,2016
6 Bahouri H,Chemin J-Y,Danchin R.Tempered distributions and Fourier transform on the Heisenberg group.ArX-iv:1705.02195,2017
7 Bahouri H,Chemin J-Y,Gallagher I.Precised Hardy inequalities on Rdand on the Heisenberg group Hd.S′eminaire′Equations aux D′eriv′ees Partielles de l’′Ecole Polytechnique(2004-2005),Expos′e XIX,2005
8 Bahouri H,Chemin J-Y,Xu C-J.Trace theorem in Sobolev spaces associated with H¨ormander’s vector fields.In:Partial Differential Equations and Their Applications.River Edge:World Sci Publ,1999,1-14
9 Bahouri H,Chemin J-Y,Xu C-J.Trace theorem on the Heisenberg group on homogeneous hypersurfaces.In:Phase Space Analysis of Partial Differential Equations.Progress in Nonlinear Differential Equations and Their Applications,vol.69.Boston:Birkh¨auser,2006,1-15
10 Bahouri H,Chemin J-Y,Xu C-J.Trace theorem on the Heisenberg group.Ann Inst Fourier(Grenoble),2009,59:491-514
11 Bahouri H,Fermanian Kammerer C,Gallagher I.Phase Space Analysis on the Heisenberg Group.Ast′erisque,vol.345.Paris:Soc Math France,2012
12 Bahouri H,Fermanian Kammerer C,Gallagher I.Dispersive estimates for the Schr¨odinger operator on step 2 stratified Lie groups.Anal PDE,2016,9:545-574
13 Bahouri H,G′erard P,Xu C-J.Espaces de Besov et estimations de Strichartz g′en′eralis′ees sur le groupe de Heisenberg.J Anal Math,2000,82:93-118
14 Beals R,Greiner P.Calculus on Heisenberg Manifolds.Annals of Mathematics Studies,vol.119.Princeton:Princeton University Press,1988
15 Christ M,Geller D,Glowacki P,et al.Pseudodifferential operators on groups with dilations.Duke Math J,1992,68:31-65
16 Colin de Verdi`ere Y,Hillairet L,Tr′elat E.Spectral asymptotics for sub-Riemannian Laplacians.I:Quantum ergodicity and quantum limits in the 3D contact case.Duke Math J,2018,167:109-174
17 Damek E,Ricci F.Harmonic analysis on solvable extensions of H-type groups.J Geom Anal,1992,2:213-248
18 Del Hierro M.Dispersive and Strichartz estimates on H-type groups.Studia Math,2005,169:1-20
19 Dixmier J.C*-Algebras.North-Holland Mathematical Library,vol.15.Amsterdam-New York-Oxford:North-Holland,1977
20 Faraut J,Harzallah K.Deux Cours d’Analyse Harmonique.Progress in Mathematics,vol.69.Boston:Birkh¨auser,1987
21 Fermanian Kammerer C,Fischer V.Defect measures on graded Lie groups.ArXiv:1707.04002,2017
22 Fermanian Kammerer C,G′erard P.A Landau-Zener formula for non-degenerated involutive codimension 3 crossings.Ann Henri Poincar′e,2003,4:513-552
23 Fischer V,Ricci F,Yakimova O.Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I:Rank-one actions on the centre.Math Z,2012,271:221-255
24 Fischer V,Ruzhansky M.Quantization on Nilpotent Lie Groups.Progress in Mathematics,vol.314.Basel:Birkh¨auser,2016
25 Folland G B,Stein E.Hardy Spaces on Homogeneous Groups.Princeton:Princeton University Press,1982
26 Geller D.Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability.Mathematical Notes,vol.37.Princeton:Princeton University Press,1990
27 G′erard P.Mesures semi-classiques et ondes de Bloch.S′eminaire′Equations aux D′eriv′ees Partielles de l’′Ecole Polytechnique(1990-1991),Expos′e XVI,1991
28 G′erard P,Leichtnam E.Ergodic properties of eigenfunctions for the Dirichlet problem.Duke Math J,1993,71:559-607
29 G′erard P,Markowich P A,Mauser N J,et al.Homogenization limits and Wigner transforms.Comm Pure Appl Math,1997,50:323-379
30 Helffer B.30 ans d’analyse semi-classique:Bibliographie comment′ee.Http://www.math.u-psud.fr/～helffer/histoire2003.ps
31 Helffer B J,Sj¨ostrand J.Equation de Schr¨odinger avec champ magn′etique et′equation de Harper.In:Lecture Notes in Physics,vol.345.Berlin:Springer,1989,118-198
32 H¨ormander L.The Weyl calculus of pseudodifferential operators.Comm Pure Appl Math,1979,32:360-444
33 Kaplan A.Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms.Trans Amer Math Soc,1980,258:147-153
34 Kor′anyi A.Some applications of Gelfand pairs in classical analysis.In:Harmonic Analysis and Group Representations.Naples:Liguori,1982,333-348
35 L′evy G.The Fourier transform on 2-step Lie groups.ArXiv:1712.09880,2017
36 Lions P-L,Paul T.Sur les mesures de Wigner.Rev Mat Iberoamericana,1993,9:553-618
37 Mantoiu M,Ruzhansky M.Pseudo-differential operators,Wigner transform and Weyl systems on type I locally compact groups.Doc Math,2017,22:1539-1592
38 Ruzhansky M,Turunen V.On the Fourier analysis of operators on the torus.In:Modern Trends in Pseudo-Differential Operators.Operator Theory:Advances and Applications,vol.172.Basel:Birkh¨auser,2007,87-105
39 Shubin M L A.Pseudodifferential Operators and Spectral Theory.Springer Series in Soviet Mathematics.Berlin:Springer-Verlag,1987
40 Taylor M E.Noncommutative Microlocal Analysis.I.Memoirs of the American Mathematical Society,vol.52.Providence:Amer Math Soc,1984
41 Wigner E O.Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra.New York:Academic Press,1959