Abstract
In this paper we shall obtain the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator. This is achieved by means of a complete description of the singularities of the Fourier transform of the spectral function for low frequencies.
This paper was written while the author was a member of the Institute for Advanced Study, Princeton, N. J.
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References
Agmon, S. und Kannai, Y, On the asymptotic behavior of spectral functions and resolvent kernels of elliptic operators. Israel J. Math., 5 (1967), 1–30.
Avakumovg, V. G., Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z., 65 (1956), 327–344.
Carleman, T., Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes. C. R. 8ème Congr. des Math. Scand. Stockholm 1934; 34–44 (Lund 1935).
Garding, L., On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differential operators. Math. Scand., 1 (1953), 237–255.
Garding, L., On the asymptotic properties of the spectral function belonging to a self-adjoint semi-bounded extension of an elliptic differential operator. Kungl. Fysiogr. Sällsk. i Lund Förh., 24, No. 21 (1954), 1–18.
Hörmander, L., Pseudo-differential operators. Comm. Pure Appl. Math., 18 (1965), 501–517.
Hörmander, L., Pseudo-differential operators and hypoelliptic equations. Amer. Math. Soc. Proc. Symp. Pure Math., 10 (1968), 138–183.
Hörmander, L., On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators. Recent Advances in the Basic Sciences, Yeshiva University Conference November 1966,155–202 (to appear).
Lax, P. D., Asymptotic solutions of oscillatory initial value problems. Duke Math. J., 24 (1957), 627–646.
Lewitan, B. M., On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order. Ízv. Akad. Nauk SSSR Sér. Mat., 16 (1952), 325–352.
Lewitan, B. M., On the asymptotic behavior of the spectral function and the eigenfunction expansion of self-adjoint differential equations of the second order II. Izv. Akad. Nauk SSSR Sér. Mat., 19 (1955), 33–58.
Minaxshisdndaram, S. und Pleijel, A., Some properties of the eigenfunction of the Laplace operator on Riemannian manifolds. Canad. J. Math. 4 (1952), 26–30.
Muller, C., Spherical harmonics. Springer-Verlag lecture notes in mathematics, 17 (1966).
Seeley, R. T., Complex powers of an elliptic operator. Amer. Math. Soc. Proc. Symp. Pure Math., 10 (1968), 288–307.
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© 1994 Springer-Verlag Berlin Heidelberg
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Hörmander, L. (1994). The Spectral Function of an Elliptic Operator. In: Brüning, J., Guillemin, V.W. (eds) Mathematics Past and Present Fourier Integral Operators. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03030-1_4
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DOI: https://doi.org/10.1007/978-3-662-03030-1_4
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