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 Uspekhi Mat. Nauk, 1971, Volume 26, Issue 6(162), Pages 151–212 (Mi umn5279)

Asymptotic behaviour of the spectral function of an elliptic equat

B. M. Levitan

Abstract: In this paper we consider the asymptotic behaviour of the spectral function of an elliptic differential (pseudodifferential) equation or system of equations.
For the case of differential operators this problem has been widely studied, and various methods have been developed for its solution (see [1] and [2] for a urvey of these methods).
We consider in this paper just one of these methods. It is based on a study of the structure of the fundamental solution of the Cauchy problem for a hyperbolic differential (pseudodifferential) equation. The method we use is called “the method of geometrical optics”. For a system of first order differential equations it was originally developed in detail by Lax [8], and for pseudodifferential equations by Hörmander [2] and independently by Eskin [17], [18] and Maslov [19].
In [2] Hörmander also investigates the asymptotic behaviour of the spectral function for an elliptic pseudodifferential first order operator. Using some important results of Seeley [5] one can then derive the asymptotic behaviour of the spectral function of an elliptic differential operator of arbitrary order.
Similar methods have previously been applied by the author in [3], [4] for second order elliptic differential operators.
In this article we give a partial account of the results of [8] and [9]. We also present some new results due to the author, concerning both the structure of the fundamental solution of the Cauchy problem and the asymptotic behaviour of the spectral function.

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English version:
Russian Mathematical Surveys, 1971, 26:6, 165–232

Bibliographic databases:

UDC: 517.5+517.9
MSC: 35B40, 35J45, 35J15, 35P05, 35Sxx, 35A08

Citation: B. M. Levitan, “Asymptotic behaviour of the spectral function of an elliptic equat”, Uspekhi Mat. Nauk, 26:6(162) (1971), 151–212; Russian Math. Surveys, 26:6 (1971), 165–232

Citation in format AMSBIB
\Bibitem{Lev71} \by B.~M.~Levitan \paper Asymptotic behaviour of the spectral function of an elliptic equat \jour Uspekhi Mat. Nauk \yr 1971 \vol 26 \issue 6(162) \pages 151--212 \mathnet{http://mi.mathnet.ru/umn5279} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=402297} \zmath{https://zbmath.org/?q=an:0236.35035} \transl \jour Russian Math. Surveys \yr 1971 \vol 26 \issue 6 \pages 165--232 \crossref{https://doi.org/10.1070/RM1971v026n06ABEH001274} 

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This publication is cited in the following articles:
1. G. A. Suvorchenkova, “O reshenii zadachi Koshi dlya lineinogo differentsialnogo uravneniya pervogo poryadka s operatornymi koeffitsientami”, UMN, 31:1(187) (1976), 263–264
2. V. Ya. Ivrii, “Accurate spectral asymptotics for elliptic operators that act in vector bundles”, Funct. Anal. Appl., 16:2 (1982), 101–108
3. B. S. Pavlov, “The theory of extensions and explicitly-soluble models”, Russian Math. Surveys, 42:6 (1987), 127–168
4. Yu. G. Safarov, “Exact asymptotics of the spectrum of a boundary value problem, and periodic billiards”, Math. USSR-Izv., 33:3 (1989), 553–573
5. Yu. G. Safarov, “Asymptotic of the spectral function of a positive elliptic operator without the nontrap condition”, Funct. Anal. Appl., 22:3 (1988), 213–223
6. A. I. Kozko, A. S. Pechentsov, “The spectral function of a singular differential operator of order $2m$”, Izv. Math., 74:6 (2010), 1205–1224
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