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1998 | 127 | 2 | 169-190
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The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin

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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 $|∂^{α}a| ≤ C_αa^{1-ϱ}$ 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.
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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.
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