Rieffel’s pseudodifferential calculus and spectral analysis of quantum Hamiltonians
Annales de l'Institut Fourier, Volume 62 (2012) no. 4, pp. 1551-1580.

We use the functorial properties of Rieffel’s pseudodifferential calculus to study families of operators associated to topological dynamical systems acted by a symplectic space. Information about the spectra and the essential spectra are extracted from the quasi-orbit structure of the dynamical system. The semi-classical behavior of the families of spectra is also studied.

On utilise les propriétés functorielles du calcul pseudodifferentiel de Rieffel pour étudier des familles d’opérateurs associés à des systèmes dynamiques topologiques sur lesquelles agit un espace symplectique. On obtient des informations sur le spectre et le spectre essentiel à partir de la structure des quasi-orbites du système dynamique. Le comportement semi-classique des familles des spectres est aussi étudié.

DOI: 10.5802/aif.2729
Classification: 35S05, 81Q10, 46L55, 47C15
Keywords: Pseudodifferential operator, essential spectrum, random operator, semiclassical limit, noncommutative dynamical system
Mot clés : Opérateur pseudodifferentiel, spectre essentiel, opérateur aléatoire, limite semiclassique, systéme dynamique non-commutative
Măntoiu, Marius 1

1 Universidad de Chile, Facultad de Ciencias, Departamento de Matemáticas, Las Palmeras 3425, Casilla 653 Santiago, Chile
@article{AIF_2012__62_4_1551_0,
     author = {M\u{a}ntoiu, Marius},
     title = {Rieffel{\textquoteright}s pseudodifferential calculus and spectral analysis of quantum {Hamiltonians}},
     journal = {Annales de l'Institut Fourier},
     pages = {1551--1580},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {4},
     year = {2012},
     doi = {10.5802/aif.2729},
     zbl = {1253.35232},
     mrnumber = {3025750},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2729/}
}
TY  - JOUR
AU  - Măntoiu, Marius
TI  - Rieffel’s pseudodifferential calculus and spectral analysis of quantum Hamiltonians
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 1551
EP  - 1580
VL  - 62
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2729/
DO  - 10.5802/aif.2729
LA  - en
ID  - AIF_2012__62_4_1551_0
ER  - 
%0 Journal Article
%A Măntoiu, Marius
%T Rieffel’s pseudodifferential calculus and spectral analysis of quantum Hamiltonians
%J Annales de l'Institut Fourier
%D 2012
%P 1551-1580
%V 62
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2729/
%R 10.5802/aif.2729
%G en
%F AIF_2012__62_4_1551_0
Măntoiu, Marius. Rieffel’s pseudodifferential calculus and spectral analysis of quantum Hamiltonians. Annales de l'Institut Fourier, Volume 62 (2012) no. 4, pp. 1551-1580. doi : 10.5802/aif.2729. http://www.numdam.org/articles/10.5802/aif.2729/

[1] Amrein, W. O.; Boutet de Monvel, A.; Georgescu, V. C 0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Birkhäuser, Basel, 1996 | MR | Zbl

[2] Athmouni, N.; Măntoiu, M.; Purice, R. On the Continuity of Spectra for Families of Magnetic Pseudodifferential Operators, J. Math. Phys., Volume 51, 083517 (2010) | MR

[3] Bellissard, J.; Herrmann, D.J.L.; Zarrouati, M. Hull of Aperiodic Solids and Gap Labelling Theorems, Directions in Mathematical Quasicrystals (CRM Monograph Series), Volume 13, 2000, pp. 207-259 | MR | Zbl

[4] Carmona, R.; Lacroix, J. Spectral Theory of Random Schrödinger Operators, Birkhäuser Boston Inc., Boston, MA, 1990 | MR | Zbl

[5] Davies, E. B. Decomposing the Essential Spectrum, J. Funct. Anal., Volume 257 (2009) no. 2, pp. 506-536 | DOI | MR | Zbl

[6] de Leeuw, K.; Mirkil, H. Translation-invariant function algebras on abelian groups, Bull. Soc. Math. France, Volume 88 (1960), pp. 345-370 | Numdam | MR | Zbl

[7] Folland, G. B. Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, Princeton, NJ, 1989 | MR | Zbl

[8] Georgescu, V. On the Structure of the Essential Spectrum of Elliptic Operators in Metric Spaces, J. Funct. Anal., Volume 220 (2011), pp. 1734-1765 | DOI | MR | Zbl

[9] Georgescu, V.; Iftimovici, A. Crossed Products of C * -Algebras and Spectral Analysis of Quantum Hamiltonians, Commun. Math. Phys., Volume 228 (2002), pp. 519-530 | DOI | MR | Zbl

[10] Georgescu, V.; Iftimovici, A. C * -Algebras of Quantum Hamiltonians, Operator Algebras and Mathematical Physics (Constanta, 2001), Theta, Bucharest, 2003, pp. 123-167 | MR | Zbl

[11] Georgescu, V.; Iftimovici, A. Localizations at Infinity and Essential Spectrum of Quantum Hamiltonians. I. General Theory, Rev. Math. Phys., Volume 18 (2006) no. 4, pp. 417-483 | DOI | MR | Zbl

[12] Helffer, B.; Mohamed, A. Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique, Ann. Inst. Fourier, Volume 38 (1988), pp. 95-112 | DOI | EuDML | Numdam | MR | Zbl

[13] Iftimie, V.; Măntoiu, M.; Purice, R. Magnetic Pseudodifferential Operators, Publ. RIMS, Volume 43 (2007) no. 3, pp. 585-623 | DOI | MR | Zbl

[14] Last, Y.; Simon, B. The Essential Spectrum of Schrödinger, Jacobi and CMV Operators, J. d’Analyse Math., Volume 98 (2006), pp. 183-220 | DOI | MR | Zbl

[15] Lauter, R.; Monthubert, B.; Nistor, V. Spectral Invariance for Certain Algebras of Pseudodifferential Operators, J. Inst. Math. Jussieu, Volume 4 (2005) no. 3, pp. 405-442 | DOI | MR | Zbl

[16] Lauter, R.; Nistor, V. Analysis of Geometric Operators on Open Manifolds: a Groupoid Approach, Quantization of Singular Symplectic Quotients (Progr. Math.), Volume 198, Birkhäuser, Basel, 2001, pp. 181-229 | MR | Zbl

[17] Lein, M.; Măntoiu, M.; Richard, S. Magnetic Pseudodifferential Operators with Coefficients in C * -algebras, Publ. RIMS Kyoto Univ., Volume 46 (2010), pp. 595-628 | MR | Zbl

[18] Măntoiu, M. Compactifications, Dynamical Systems at Infinity and the Essential Spectrum of Generalized Schödinger Operators, J. reine angew. Math., Volume 500 (2002), pp. 211-229 | DOI | MR | Zbl

[19] Măntoiu, M. On Abelian C * -Algebras that are Independent with Respect to a Filter, J. London Math. Soc., Volume 71 (2005) no. 3, pp. 740-758 | DOI | MR | Zbl

[20] Măntoiu, M.; Purice, R. The Magnetic Weyl Calculus, J. Math. Phys., Volume 45 (2004) no. 4, pp. 1394-1417 | DOI | MR | Zbl

[21] Măntoiu, M.; Purice, R.; Richard, S. Spectral and Propagation Results for Magnetic Schrödinger Operators; a C * -Algebraic Framework, J. Funct. Anal., Volume 250 (2007), pp. 42-67 | DOI | MR | Zbl

[22] Pastur, L. A.; Figotin, A. Spectra of Random and Almost Periodic Operators, Springer Verlag, Berlin, 1992 | MR | Zbl

[23] Rabinovich, V. S.; Roch, S.; Roe, J. Fredholm Indices of Band-Dominated Operators, Int. Eq. Op. Theory, Volume 49 (2004), pp. 221-238 | DOI | MR | Zbl

[24] Rabinovich, V. S.; Roch, S.; Silbermann, B. Limit Operators and their Applications in Operator Theory, Operator Theory: Advances and Applications, 150, Birkhäuser, Basel, 2004 | MR | Zbl

[25] Reed, M.; Simon, B. Methods of Modern Mathematical Physics I, Functional Analysis, Academic Press Inc., [Harcourt Brace Jovanovich Publishers], New York, second edition, 1980 | MR | Zbl

[26] Rieffel, M. A. Quantization and C * -Algebras, Doran R. S. (ed.) C * -Algebras: 1943–1993 (Contemp. Math.), Volume 167, AMS Providence, pp. 67-97 | MR | Zbl

[27] Rieffel, M. A. Deformation Quantization for Actions of d , 506, Mem. AMS, 1993 | MR | Zbl

Cited by Sources: