Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the index and spectrum of differential operators on $ \mathbb{R}^{N}$


Author: Patrick J. Rabier
Journal: Proc. Amer. Math. Soc. 135 (2007), 3875-3885
MSC (2000): Primary 47A53, 47F05, 35J45
DOI: https://doi.org/10.1090/S0002-9939-07-08896-X
Published electronically: August 29, 2007
MathSciNet review: 2341938
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $ P(x,\partial )$ is an $ r\times r$ system of differential operators on $ \mathbb{R}^{N}$ having continuous coefficients with vanishing oscillation at infinity, the Cordes-Illner theory ensures that $ P(x,\partial )$ is Fredholm from $ (W^{m,p})^{r}$ to $ (L^{p})^{r}$ for all or no value $ p\in (1,\infty ).$ We prove that both the index (when defined) and the spectrum of $ P(x,\partial )$ are independent of $ p.$


References [Enhancements On Off] (What's this?)

  • 1. Bott, R. and Seeley, R., Some remarks on the paper of Callias: ``Axial anomalies and index theorems on open spaces'' [Comm. Math. Phys. 62 (1978) 213-234 ], Comm. Math. Phys. 62 (1978) 235-245. MR 507781 (80h:58045b)
  • 2. Cordes, H. O., Beispiele von Pseudo-Differentialoperator-Algebren, Applicable Anal. 2 (1972) 115-129. MR 0402541 (53:6360)
  • 3. Cordes, H. O., On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal. 18 (1975) 115-131. MR 0377599 (51:13770)
  • 4. Davies, E. B., $ L^{p}$ spectral theory of higher-order elliptic differential operators, Bull. London Math. Soc. 29 (1997) 513-546. MR 1458713 (98d:35164)
  • 5. Dieudonné, J., Éléments d'analyse, Vol. 9, Gauthier-Villars, Paris, 1982. MR 658305 (84a:57021)
  • 6. Fan, Q. and Wong, M. W., A characterization of Fredholm pseudo-differential operators, J. London Math. Soc. (2) 55 (1997) 139-145. MR 1423291 (97j:47073)
  • 7. Fedosov, B. V., Analytical formulas for the index of elliptic operators, Trans. Mosc. Math. Soc. 30 (1974) 159-240. MR 0420731 (54:8743)
  • 8. Geymonat, G., Sui problemi ai limiti per i sistemi lineari ellittici, Ann. Mat. Pura Appl. 69 (1965) 207-284. MR 0196262 (33:4454)
  • 9. Hempel, R. and Voigt, J., The spectrum of a Schrödinger operator in $ L^{p}(\mathbb{R}^{\nu })$ is independent of $ p,$ Comm. Math. Phys. 104 (1986), 243-250. MR 836002 (87h:35247)
  • 10. Hieber, M. and Schrohe, E., $ L^{p}$ spectral independence of elliptic operators via commutator estimates, Positivity 3 (1999) 259-272. MR 1708648 (2000e:35034)
  • 11. Hörmander, L., The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979) 360-444. MR 517939 (80j:47060)
  • 12. Illner, R., On algebras of pseudo differential operators in $ L^{p}(\mathbb{R}^{n}),$ Comm. Partial Differ. Eq. 2 (1977) 359-393. MR 0442758 (56:1137b)
  • 13. Kozhevnikov, A., Complete scale of isomorphisms for elliptic pseudodifferential boundary-value problems, J. London Math. Soc. 64 (2001) 409-422. MR 1853460 (2002j:35333)
  • 14. Leopold, H.-G. and Schrohe, E., Invariance of the $ L_{p}$ spectrum for hypoelliptic operators, Proc. Amer. Math. Soc., 125 (1997) 3679-3687. MR 1423315 (98b:35125)
  • 15. McOwen, R. C., The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math. 32 (1979) 783-795. MR 539158 (81m:47069)
  • 16. Rabier, P. J., Fredholm operators, semigroups and the asymptotic and boundary behavior of solutions of PDEs, J. Differ. Eq. 193 (2003)460-480. MR 1998964 (2004f:47059); Corrigendum, J. Differ. Eq. 237 (2007) 257.
  • 17. Rabier, P. J. and Stuart, C. A., Fredholm properties of Schrödinger operators in $ L^{p}(\mathbb{R}^{N}),$ Differ. Integral Eq. 13 (2000) 1429-1444. MR 1787075 (2001m:47103)
  • 18. Seeley, R. T., The index of elliptic systems of singular integral operators, J. Math. Anal Appl., 7 (1963) 289-309. MR 0159247 (28:2464)
  • 19. Sun, S. H., A Banach algebra approach to the Fredholm theory of pseudodifferential operators, Sci. Sinica, Series A, 27 (1984) 337-344. MR 763970 (86b:47092)
  • 20. Taylor, M. E., Gelfand theory of pseudo differential operators and hypoelliptic operators, Trans. Amer. Math. Soc. 153 (1971), 495-510. MR 0415430 (54:3517)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A53, 47F05, 35J45

Retrieve articles in all journals with MSC (2000): 47A53, 47F05, 35J45


Additional Information

Patrick J. Rabier
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: rabier@imap.pitt.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08896-X
Keywords: Fredholm operator, index, differential operator, system, spectrum.
Received by editor(s): January 14, 2006
Received by editor(s) in revised form: August 27, 2006
Published electronically: August 29, 2007
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society