# 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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topGłowacki, 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

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

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