We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Advertisement

Localization results for zero order pseudodifferential operators

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

  • 104 Accesses

Abstract

We show phase space localization at suitable energies for zero order pseudodifferential operators, implying non-propagation properties for the associated evolution groups.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

References

  1. 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)

  2. 2.

    Amrein, W.O., Măntoiu, M., Purice, R.: Propagation properties for Schrödinger operators affiliated with certain \(C^*\)-algebras. Ann. H. Poincaré 3(6), 1215–1232 (2002)

  3. 3.

    Beltiţă, I., Măntoiu, M.: Rieffel Deformation and Crossed Products. Int. Math. Res. Not. (2013)

  4. 4.

    Chandler-Wilde, S.N., Lindner, M.: Limit operators, collective compactness, and the spectral theory of infinite matrices. Mem. AMS, 210, 111 p. (2011)

  5. 5.

    Davies, E.B.: Decomposing the essential spectrum. J. Funct. Anal. 257(2), 506–536 (2009)

  6. 6.

    Davies, E.B., Simon, B.: Scattering theory for systens with different spatial asymptotics on the left and right. Commun. Math. Phys. 63, 277–301 (1978)

  7. 7.

    Folland, G.B.: Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)

  8. 8.

    Georgescu, V.: On the structure of the essential spectrum of elliptic operators in metric spaces. J. Funct. Anal. 220, 1734–1765 (2011)

  9. 9.

    Georgescu, V., Iftimovici, A.: Crossed products of \(C^*\)-algebras and spectral analysis of quantum hamiltonians. Commun. Math. Phys. 228, 519–560 (2002)

  10. 10.

    Georgescu, V., Iftimovici, A.: \(C^*\)-algebras of quantum hamiltonians. In: Operator Algebras and Mathematical Physics (Constanta, 2001), pp. 123–167. Theta, Bucharest (2003)

  11. 11.

    Georgescu, V., Iftimovici, A.: Localizations at infinity and essential spectrum of quantum hamiltonians. I. General theory. Rev. Math. Phys. 18(4), 417–483 (2006)

  12. 12.

    Helffer, B., Mohamed, A.: Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique. Ann. Inst. Fourier 38, 95–112 (1988)

  13. 13.

    Lange, B.V., Rabinovich, V.S.: Pseudodifferential operators on \(\mathbb{R}^n\) and limit operators. Mat. Sb. (N.S.), 129(171)(2), 175–185 (1986)

  14. 14.

    Lein, M., Măntoiu, M., Richard, S.: Magnetic pseudodifferential operators with coefficients in \(C^*\)-algebras. Publ. RIMS Kyoto Univ. 46, 595–628 (2010)

  15. 15.

    Last, Y., Simon, B.: The essential spectrum of Schrödinger, Jacobi and CMV operators. J. d’Analyse Math. 98, 183–220 (2006)

  16. 16.

    Lindner, M.: Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method. Frontiers in Mathematics, Birkhäuser (2006)

  17. 17.

    Măntoiu, M.: Compactifications, dynamical systems at infinity and the essential spectrum of generalized Schödinger operators. J. Reine Angew. Math. 550, 211–229 (2002)

  18. 18.

    Măntoiu, M.: Rieffel’s pseudodifferential calculus and spectral analysis for quantum Hamiltonians. Ann. Inst. Fourier 62(4), 1551–1558 (2012)

  19. 19.

    Măntoiu, M.: On the essential spectrum of phase-space anisotropic pseudodifferential operators. Math. Proc. Camb. Philos. Soc. 154(1), 29–39 (2013)

  20. 20.

    Măntoiu, M., Purice, R., Richard, S.: Spectral and propagation results for magnetic Schrödinger operators; a \(C^*\)-algebraic framework. J. Funct. Anal. 250, 42–67 (2007)

  21. 21.

    Rabinovich, V.S.: Essential spectrum of perturbed pseudodifferential operators. Applications to the Schrödinger, Klein-Gordon and Dirac operators. Russ. J. Math. Phys. 12(1), 62–80 (2005)

  22. 22.

    Rabinovich, V.S., Roch, S.: The essential spectrum of Schrödinger operators on lattices. J. Math. Phys. A Math. Gen. 39, 8377–8394 (2006)

  23. 23.

    Rabinovich, V.S., Roch, S., Roe, J.: Fredholm indices of band-dominated operators. Int. Equ. Oper. Theory 49, 221–238 (2004)

  24. 24.

    Rabinovich, V.S., Roch, S., Silbermann, B.: Fredholm theory and finite section method for band-dominated operators. Integral Equ. Oper. Theory 40(3), 342–381 (2001)

  25. 25.

    Rabinovich, V.S., Roch, S., Silbermann, B.: Limit Operators and their Applications in Operator Theory, Operator Theory: Advances and Applications, vol. 150. Birkhäuser, Basel (2004)

  26. 26.

    Rieffel, M.A.: Deformation Quantization for Actions of \(\,\mathbb{R}^d\). Mem. AMS, 506 (1993)

Download references

Author information

Correspondence to M. Măntoiu.

Additional information

This work was completed with the support of M. Măntoiu’s Fondecyt Project 1120300.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

García, J., Măntoiu, M. Localization results for zero order pseudodifferential operators. J. Pseudo-Differ. Oper. Appl. 5, 255–276 (2014). https://doi.org/10.1007/s11868-013-0084-y

Download citation

Keywords

  • Pseudodifferential operator
  • Spectrum
  • Rieffel quantization
  • \(C^*\)-algebra
  • Propagation

Mathematics Subject Classification (2010)

  • Primary 35S05
  • 46L55
  • Secondary 47C15
  • 81Q10