# The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin

Studia Mathematica (1998)

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

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## 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 $|{\partial }^{\alpha }a|\le {C}_{\alpha }{a}^{1-\varrho }$ for some ϱ>0 and all |α|>0, ${\partial }^{\alpha }a$ is bounded for large |α|, and $li{m}_{w\to \infty }a\left(w\right)=\infty$. 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
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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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