This talk will describe some results on the inverse spectral problem on a compact riemannian manifold (possibly with boundary) which are based on V. Guillemin's strategy of normal forms. It consists of three steps : first, put the wave group into a normal form around each closed geodesic. Second, determine the normal form from the spectrum of the laplacian. Third, determine the metric from the normal form. We will try to explain all three steps and to illustrate with simple examples such as surfaces of revolution.
@article{JEDP_1998____A15_0,
author = {Zelditch, Steven},
title = {Normal form of the wave group and inverse spectral theory},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
year = {1998},
pages = {1-18},
mrnumber = {99h:58197},
zbl = {01808724},
language = {en},
url = {http://dml.mathdoc.fr/item/JEDP_1998____A15_0}
}
Zelditch, Steve. Normal form of the wave group and inverse spectral theory. Journées équations aux dérivées partielles, (1998), pp. 1-18. http://gdmltest.u-ga.fr/item/JEDP_1998____A15_0/
[B.B] , , Short-Wavelength Diffraction Theory, Springer Series on Wave Phenomena 4, Springer-Verlag, New York (1991) | MR 94f:78004 | Zbl 0742.35002
[CV.1] , Sur les longueurs des trajectoires périodiques d'un billard, In : P. Dazord and N. Desolneux-Moulis (eds.) Géométrie Symplectique et de Contact : Autour du Théorème de Poincaré-Birkhoff. Travaux en Cours, Sém. Sud-Rhodanien de Géométrie III Paris : Herman (1984), 122-139. | MR 86a:58078 | Zbl 0599.58039
[CV.2] , Spectre conjoint d'opérateurs pseudo-différentiels qui commutent II. Le cas intégrable, Math.Zeit. 171 (1980), 51-73. | MR 81i:58046 | Zbl 0478.35073
[CV.3] , Quasi-modes sur les variétés Riemanniennes, Inv. Math 43 (1977), 15-52. | MR 58 #18615 | Zbl 0449.53040
[D.G] and , The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. Math. 24 (1975), 39-80. | MR 53 #9307 | Zbl 0307.35071
[F.Z] and , Inverse spectral problem for surfaces of revolution II (in preparation).
[F.G] and , On the period spectrum of a symplectic mapping, J. Fun. Anal. 100, (1991) 317-358. | MR 92j:58083 | Zbl 0739.58020
[Go] , CBMS Lectures (1996).
[G.1] , Wave trace invariants, Duke Math. J. 83 (1996), 287-352. | MR 97f:58131 | Zbl 0858.58051
[G.2] , Wave-trace invariants and a theorem of Zelditch, Duke Int.Math.Res.Not. 12 (1993), 303-308. | MR 95f:58077 | Zbl 0798.58073
[G.M] and , The Poisson summation formula for manifolds with boundary, Adv. in Math.32 (1979), 204-232. | MR 80j:58066 | Zbl 0421.35082
[HoI-IV] , Theory of Linear Partial Differential Operators I-IV, Springer-Verlag, New York (1985).
[Kac] , On applying mathematics : reflections and examples, in Mark Kack : Probability, Number Theory, and Statistical Physics, K.Baclawski and M.D.Donskder (eds.), MIT Press, Cambridge (1979). | Zbl 0275.00009
[M.M] and , Spectral invariants of convex planar regions, J. Diff.Geom. 17 (1982), 475-502. | MR 85d:58084 | Zbl 0492.53033
[M] , The wave equation for a hypoelliptic operator with symplectic characteristics of codimension two, Journal D'Analyse Math. XLIV (1984/1985), 134-182. | MR 87e:58199 | Zbl 0599.35139
[P] , Length spectrum invariants of Riemannian manifolds, Math.Zeit. 213 (1993), 311-351. | MR 94g:58174 | Zbl 0804.53068
[Sj] , Semi-excited states in nondegenerate potential wells, Asym.An.6 (1992) 29-43. | MR 93m:35052 | Zbl 0782.35050
[W] , Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 (1977), 883-892. | MR 58 #2919 | Zbl 0385.58013
[Z.1] , Wave invariants at elliptic closed geodesics, Geom.Anal.Fun.Anal. 7 (1997), 145-213. | MR 98f:58191 | Zbl 0876.58010
[Z.2] , Wave invariants for non-degenerate closed geodesics, Geom.Anal.Fun.Anal. 8 (1998), 179-217. | MR 98m:58136 | Zbl 0908.58022
[Z.3] , Inverse spectral problem for surfaces of revolution (to appear in J.Diff.Geom.). | Zbl 0938.58027