# Spectral Asymptotics

• Lars Hörmander
Part of the Grundlehren der mathematischen Wissenschaften book series (CLASSICS)

## Summary

This chapter is devoted to the asymptotic properties of the eigenvalues and the spectral function for self-adjoint elliptic operators. In Sections 29.1 and 29.2 we study operators on a compact manifold without boundary. If P is positive and of order m then P1/m is a pseudo-differential operator with the eigenfunctions for the eigenvalue λ equal to those of λ m We shall therefore generalize the problem by studying first order pseudo-differential operators. This is actually a simplification from the geometrical as well as from the analytical point of view. In fact, as emphasized in Chapter XXVI, the Hamilton flow defined by the principal symbol is a flow on the cosphere bundle when the order is one. Moreover, the corresponding unitary group te -itP is the solution operator for the hyperbolic equation (D t +P)u=0 when the order of P is one. This operator was studied by means of energy estimates in Section 23.1. Here we identify the solution operator as a Fourier integral operator, which we actually know from Chapter XXVI also. This gives a perfect substitute for the Hadamard construction used in Section 17.5 and allows us to determine the singularities of the kernel of e -itP for any t. As a result we obtain not only an analogue of the asymptotic formulas proved there in the second order case but also an improved formula with a second term if the set of closed orbits of the Hamilton field is of measure 0. In Section 29.3 we give a similar improvement for the Dirichlet problem for a second order elliptic operator in a manifold with boundary.

Section 29.2 is devoted to the opposite case where all bicharacteristics are closed with the same period Then the eigenvalues tend to cluster around an arithmetic sequence with difference 2π/ determined by a Maslov class and spread out in a manner determined by the subprincipal symbol. These results illuminate the example given in Section 17.5 where we used spherical harmonics to study the Laplacean on the unit sphere.

## Notes

1. [22]
Hörmander, L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968).
2. [2]
Chazarain, J.: Formules de Poisson pour les variétés riemanniennes. Invent. Math. 24, 65–82 (1974).
3. [1]
Duistermaat, J.J. and V.W. Guillemin: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent Math. 29, 39–79 (1975).
4. [26]
Hörmander, L.: Fourier integral operators I. Acta Math. 127, 79–183 (1971).
5. [1]
Duistermaat, J.J. and V.W. Guillemin: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent Math. 29, 39–79 (1975).
6. [3]
Ivrii, Via: On the second term in the spectral asymptotics for the Laplace-Beltrami operator on a manifold with boundary. Funkcional. Anal. i Priloien. 14:2, 25-34 (1980) (Russian); also in Functional Anal. Appl. 14, 98–106 (1980).Google Scholar
7. [1]
Szegö, G.: Beiträge zur Theorie der Toeplitzschen Formen. Math. Z. 6, 167–202 (1920).
8. [2]
Guillemin, V.: Some classical theorems in spectral theory revisited. Seminar on sing. of sol. of diff. eq., Princeton University Press, Princeton, NJ., 219–259 (1979).Google Scholar
9. [1]
Widom, H.: Eigenvalué distribution in certain homogeneous spaces. J. Functional Analysis 32, 139–147 (1979).
10. [5]
Seeley, RT.: Elliptic singular integral equations. Amer Math. Soc. Symp. on Singular Integrals, 308–315 (1966).Google Scholar
11. [4]
Taylor, M.:. Pseudodifferential operators. Princeton Univ. Press, Princeton, N.J., 1981. Thorin, 0.: An extension of a convexity theorem due to M. Riesz. Kungl Fys. Sällsk. Lund. Förh. 8, No 14 (1939).Google Scholar
12. [2]
Chazarain, J.: Formules de Poisson pour les variétés riemanniennes. Invent. Math. 24, 65–82 (1974).
13. [1]
Duistermaat, J.J. and V.W. Guillemin: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent Math. 29, 39–79 (1975).
14. [I]
Cohn de Verdière. Y.: Sur le spectre des opérateurs elliptiques à bicharactéristiques toutes périodiques. Comment. Math. Helv. 54, 508–522 (1979).
15. [1]
Boutet de Monvel, L. and V. Guillemin: The spectral theory of Toeplitz operators. Ann. of Math. Studies 99 (1981).Google Scholar
16. [3]
Guillemin, V.: Some spectral results for the Laplace operator with potential on the n-sphere. Advances in Math. 27, 273–286 (1978).
17. [I]
Cohn de Verdière. Y.: Sur le spectre des opérateurs elliptiques à bicharactéristiques toutes périodiques. Comment. Math. Helv. 54, 508–522 (1979).
18. [2]
Weinstein, A.: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44, 883–892 (1977).
19. [1]
Guillemin, V.: The Radon transform on Zoll surfaces. Advances in Math. 22, 85119 (1976).