The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin

Paweł Głowacki

Studia Mathematica (1998)

  • Volume: 127, Issue: 2, page 169-190
  • ISSN: 0039-3223

Abstract

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Let A be a pseudodifferential operator on ℝ N whose Weyl symbol a is a strictly positive smooth function on W = ℝ N × ℝ N such that | ∂ α a | ≤ C α a 1 - ϱ for some ϱ>0 and all |α|>0, ∂ α a is bounded for large |α|, and l i m w → ∞ a ( w ) = ∞ . Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.

How to cite

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Głowacki, Paweł. "The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin." Studia Mathematica 127.2 (1998): 169-190. <http://eudml.org/doc/216465>.

@article{Głowacki1998,
abstract = {Let A be a pseudodifferential operator on $ℝ^N$ whose Weyl symbol a is a strictly positive smooth function on $W = ℝ^N × ℝ^N$ such that $|∂^\{α\}a| ≤ C_αa^\{1-ϱ\}$ for some ϱ>0 and all |α|>0, $∂^\{α\}a$ is bounded for large |α|, and $lim_\{w→∞\}a(w) = ∞$. Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.},
author = {Głowacki, Paweł},
journal = {Studia Mathematica},
keywords = {pseudodifferential operator; Weyl symbol; Weyl asymptotic formula; distribution of the eigenvalues; approximate spectral projectors},
language = {eng},
number = {2},
pages = {169-190},
title = {The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin},
url = {http://eudml.org/doc/216465},
volume = {127},
year = {1998},
}

TY - JOUR
AU - Głowacki, Paweł
TI - The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin
JO - Studia Mathematica
PY - 1998
VL - 127
IS - 2
SP - 169
EP - 190
AB - Let A be a pseudodifferential operator on $ℝ^N$ whose Weyl symbol a is a strictly positive smooth function on $W = ℝ^N × ℝ^N$ such that $|∂^{α}a| ≤ C_αa^{1-ϱ}$ for some ϱ>0 and all |α|>0, $∂^{α}a$ is bounded for large |α|, and $lim_{w→∞}a(w) = ∞$. Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.
LA - eng
KW - pseudodifferential operator; Weyl symbol; Weyl asymptotic formula; distribution of the eigenvalues; approximate spectral projectors
UR - http://eudml.org/doc/216465
ER -

References

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  1. [1] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer, Berlin, 1975. Zbl0308.31001
  2. [2] W. Czaja and Z. Rzeszotnik, Two remarks on spectral asymptotics for pseudodifferential operators, to appear. 
  3. [3] G. B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, N.J., 1989. Zbl0682.43001
  4. [4] P. Głowacki, Stable semigroups of measures on the Heisenberg group, Studia Math. 89 (1984), 105-138. Zbl0563.43002
  5. [5] L. Hörmander, On the asymptotic distribution of the eigenvalues of pseudodifferential operators in ℝ n , Ark. Mat. 17 (1979), 297-313. Zbl0436.35064
  6. [6] L. Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math 32 (1979), 359-443. Zbl0388.47032
  7. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators II, Springer, Berlin, 1983. Zbl0521.35002
  8. [8] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, Berlin, 1983. Zbl0521.35002
  9. [9] R. Howe, Quantum mechanics and partial differential operators, J. Funct. Anal. 38 (1980), 188-254. Zbl0449.35002
  10. [10] D. Manchon, Formule de Weyl pour les groupes de Lie nilpotents, J. Reine Angew. Math. 418 (1991), 77-129. Zbl0721.22004
  11. [11] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. Zbl0308.47002
  12. [12] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer, Berlin, 1987. 
  13. [13] V. N. Tulovskiĭ and M. A. Shubin, On the asymptotic distribution of the eigenvalues of pseudodifferential operators in ℝ n , Math. USSR-Sb. 21 (1973), 565-583. Zbl0295.35068

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