The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin

• Autorzy: Paweł Głowacki
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47G30: Pseudodifferential operators
• 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 127
• Numer: 2
• Strony: 169-190

Daty

wydano

1998

otrzymano

1997-02-10

Twórcy

autor

Paweł Głowacki

• Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Bibliografia

• [1] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer, Berlin, 1975.
• [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.
• [4] P. Głowacki, Stable semigroups of measures on the Heisenberg group, Studia Math. 89 (1984), 105-138.
• [5] L. Hörmander, On the asymptotic distribution of the eigenvalues of pseudodifferential operators in $ℝ^n$, Ark. Mat. 17 (1979), 297-313.
• [6] L. Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math 32 (1979), 359-443.
• [7] L. Hörmander, The Analysis of Linear Partial Differential Operators II, Springer, Berlin, 1983.
• [8] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, Berlin, 1983.
• [9] R. Howe, Quantum mechanics and partial differential operators, J. Funct. Anal. 38 (1980), 188-254.
• [10] D. Manchon, Formule de Weyl pour les groupes de Lie nilpotents, J. Reine Angew. Math. 418 (1991), 77-129.
• [11] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.
• [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 $ℝ^n$, Math. USSR-Sb. 21 (1973), 565-583.