Abstract and Applied Analysis

On the Spectral Asymptotics of Operators on Manifolds with Ends

Sandro Coriasco and Lidia Maniccia

Full-text: Open access


We deal with the asymptotic behaviour, for λ + , of the counting function N P ( λ ) of certain positive self-adjoint operators P with double order ( m , μ ) , m , μ > 0 ,   m μ , defined on a manifold with ends M. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined on n . By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for N P ( λ ) and show how their behaviour depends on the ratio m / μ and the dimension of M.

Article information

Abstr. Appl. Anal., Volume 2013 (2013), Article ID 909782, 21 pages.

First available in Project Euclid: 27 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Coriasco, Sandro; Maniccia, Lidia. On the Spectral Asymptotics of Operators on Manifolds with Ends. Abstr. Appl. Anal. 2013 (2013), Article ID 909782, 21 pages. doi:10.1155/2013/909782. https://projecteuclid.org/euclid.aaa/1393511841

Export citation


  • H. O. Cordes, The Technique of Pseudodifferential Operators, vol. 202, Cambridge University Press, Cambridge, Mass, USA, 1995.
  • C. Parenti, “Operatori pseudo-differenziali in $\mathbb{R}$$^{n}$ e applicazioni,” Annali di Matematica Pura ed Applicata, vol. 93, pp. 359–389, 1972.
  • R. B. Melrose, Geometric Scattering Theory, Stanford Lectures, Cambridge University Press, Cambridge, Mass, USA, 1995.
  • M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics, Springer, Berlin, Germany, 1987.
  • E. Schrohe, “Spaces of weighted symbols and weighted sobolev spaces on manifolds,” in Pseudodifferential Operators (Oberwolfach, 1986), vol. 1256 of Lecture Notes in Mathematics, pp. 360–377, Springer, Berlin, Germany, 1987.
  • Y. V. Egorov and B.-W. Schulze, Pseudo-differential operators, singularities, applications, vol. 93, Birkhäuser Verlag, Basel, Switzerland, 1997.
  • L. Maniccia and P. Panarese, “Eigenvalue asymptotics for a class of md-elliptic $\psi $do's on manifolds with cylindrical exits,” Annali di Matematica Pura ed Applicata, vol. 181, no. 3, pp. 283–308, 2002.
  • V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer, Berlin, Germany, 1998.
  • L. Hörmander, “The spectral function of an elliptic operator,” Acta Mathematica, vol. 121, no. 1, pp. 193–218, 1968.
  • V. Guillemin, “A new proof of Weyl's formula on the asymptotic distribution of eigenvalues,” Advances in Mathematics, vol. 55, no. 2, pp. 131–160, 1985.
  • H. Kumano-go, Pseudodifferential Operators, MIT Press, Boston, Mass, USA, 1981.
  • P. Boggiatto, E. Buzano, and L. Rodino, Global Hypoellipticity and Spectral Theory, vol. 92, Akademie Verlag, Berlin, Germany, 1996.
  • B. Helffer, Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques, vol. 112, Astérisque, 1984.
  • L. Hörmander, “On the asymptotic distribution of the eigenvalues of pseudodifferential operators in $\mathbb{R}$$^{n}$,” Arkiv för Matematik, vol. 17, no. 2, pp. 297–313, 1979.
  • A. Mohammed, “Comportement asymptotique, avec estimation du reste, des valor propres d'une classe d'operateurs pseudo-differentiels sur $\mathbb{R}$$^{n}$,” Mathematische Nachrichten, vol. 14, pp. 127–186, 1989.
  • F. Nicola, “Trace functionals for a class of pseudo-differential operators in $\mathbb{R}$$^{n}$,” Mathematical Physics, Analysis and Geometry, vol. 6, no. 1, pp. 89–105, 2003.
  • T. Christiansen and M. Zworski, “Spectral asymptotics for manifolds with cylindrical ends,” Annales de l'Institut Fourier, vol. 45, no. 1, pp. 251–263, 1995.
  • U. Battisti and S. Coriasco, “Wodzicki residue for operators on manifolds with cylindrical ends,” Annals of Global Analysis and Geometry, vol. 40, no. 2, pp. 223–249, 2011.
  • L. Maniccia, E. Schrohe, and J. Seiler, “Complex powers of classical SG-pseudodifferential operators,” Annali dell'Universitá di Ferrara, vol. 52, no. 2, pp. 353–369, 2006.
  • M. Borsero and S. Coriasco, “Eigenvalue asymptotics of Schrödinger-type operators on manifolds with ends,” In Preparation.
  • M. Borsero, Microlocal analysis and spectral theory of elliptic operators on non-compact manifolds [thesis], Tesi di Laurea Magistrale in Matematica, Universitá di Torino, 2011.
  • S. Coriasco, “Fourier integral operators in SG classes. I. Composition theorems and action on SG sobolev spaces,” Rendiconti del Seminario Matematico. Università e Politecnico di Torino, vol. 57, no. 4, pp. 249–302, 1999.
  • S. Coriasco, “Fourier integral operators in SG classes. II. Application to SG hyperbolic cauchy problems,” Annali dell'Università di Ferrara, vol. 44, pp. 81–122, 1998.
  • S. Coriasco and L. Rodino, “Cauchy problem for SG-hyperbolic equations with constant multiplicities,” Ricerche di Matematica, vol. 48, Supplement, pp. 25–43, 1999.
  • A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators, vol. 196, Cambridge University Press, Cambridge, Mass, USA, 1994.
  • B. Helffer and D. Robert, “Propriétés asymptotiques du spectre d'opérateurs pseudodifférentiels sur $\mathbb{R}$$^{n}$,” Communications in Partial Differential Equations, vol. 7, no. 7, pp. 795–882, 1982.
  • H. Tamura, “Asymptotic formulas with sharp remainder estimates for eigenvalues of elliptic operators of second order,” Duke Mathematical Journal, vol. 49, no. 1, pp. 87–119, 1982.
  • S. Coriasco and P. Panarese, “Fourier integral operators defined by classical symbols with exit behaviour,” Mathematische Nachrichten, vol. 242, pp. 61–78, 2002.
  • S. Coriasco and L. Maniccia, “Wave front set at infinity and hyperbolic linear operators with multiple characteristics,” Annals of Global Analysis and Geometry, vol. 24, no. 4, pp. 375–400, 2003.
  • J. J. Duistermaat, Fourier Integral Operators, vol. 130 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 1996.
  • L. H. Hörmander, The Analysis of Linear Partial Differential Operators I-IV. Classics in Mathematics, Springer, Berlin, Germany, 2009.
  • B. Helffer and D. Robert, “Comportement asymptotique précisé du spectre d'opérateurs globalement elliptiques dans $\mathbb{R}$$^{n}$,” Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, vol. 292, no. 6, pp. 363–366, 1981.
  • F. Nicola and L. Rodino, “SG pseudo-differential operators and weak hyperbolicity,” Pliska Studia Mathematica Bulgarica, vol. 15, pp. 5–20, 2003.