Two remarks about spectral asymptotics of pseudodifferential operators

Wojciech Czaja; Ziemowit Rzeszotnik

Colloquium Mathematicae (1999)

  • Volume: 80, Issue: 1, page 131-145
  • ISSN: 0010-1354

Abstract

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In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin and Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics.

How to cite

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Czaja, Wojciech, and Rzeszotnik, Ziemowit. "Two remarks about spectral asymptotics of pseudodifferential operators." Colloquium Mathematicae 80.1 (1999): 131-145. <http://eudml.org/doc/210699>.

@article{Czaja1999,
abstract = {In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin and Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics.},
author = {Czaja, Wojciech, Rzeszotnik, Ziemowit},
journal = {Colloquium Mathematicae},
keywords = {quasi-classical asymptotics; Weyl pseudo-differential operators; Weyl asymptotic formula},
language = {eng},
number = {1},
pages = {131-145},
title = {Two remarks about spectral asymptotics of pseudodifferential operators},
url = {http://eudml.org/doc/210699},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Czaja, Wojciech
AU - Rzeszotnik, Ziemowit
TI - Two remarks about spectral asymptotics of pseudodifferential operators
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 1
SP - 131
EP - 145
AB - In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin and Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics.
LA - eng
KW - quasi-classical asymptotics; Weyl pseudo-differential operators; Weyl asymptotic formula
UR - http://eudml.org/doc/210699
ER -

References

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  1. [1] R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45-57. Zbl0353.35088
  2. [2] G. B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, N.J., 1989. Zbl0682.43001
  3. [3] P. Głowacki, Stable semi-groups of measures on the Heisenberg group, Studia Math. 79 (1984), 105-138. Zbl0563.43002
  4. [4] P. Głowacki, The Weyl asymptotic formula for a class of pseudodifferential operators, ibid. 127 (1998), 169-170. 
  5. [5] L. Hörmander, On the asymptotic distribution of the pseudodifferential operators in ℝ n , Ark. Mat. 17 (1979), 297-313. Zbl0436.35064
  6. [6] L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer, Berlin, 1983. Zbl0521.35002
  7. [7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. 
  8. [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV, Academic Press, New York, 1978. Zbl0401.47001
  9. [9] V. L. Roĭtburd, The quasiclassical asymptotic behavior of the spectrum of a pseudodifferential operator, Uspekhi Mat. Nauk 31 (1976), no. 4, 275-276 (in Russian). 
  10. [10] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Nauka, Moscow, 1978 (in Russian). Zbl0451.47064
  11. [11] M. A. Shubin and V. N. Tulovskiĭ, On the asymptotic distribution of the eigenvalues of pseudodifferential operators in ℝ n , Mat. Sb. 92 (134) (1973), 571-588 (in Russian). Zbl0286.35059

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