Normal form of the wave group and inverse spectral theory
Journées équations aux dérivées partielles (1998), article no. 15, 18 p.

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.

@incollection{JEDP_1998____A15_0,
     author = {Steve Zelditch},
     title = {Normal form of the wave group and inverse spectral theory},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {15},
     pages = {1--18},
     publisher = {Universit\'e de Nantes},
     year = {1998},
     zbl = {01808724},
     mrnumber = {99h:58197},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/item/JEDP_1998____A15_0/}
}
TY  - JOUR
AU  - Steve Zelditch
TI  - Normal form of the wave group and inverse spectral theory
JO  - Journées équations aux dérivées partielles
PY  - 1998
SP  - 1
EP  - 18
PB  - Université de Nantes
UR  - https://proceedings.centre-mersenne.org/item/JEDP_1998____A15_0/
LA  - en
ID  - JEDP_1998____A15_0
ER  - 
%0 Journal Article
%A Steve Zelditch
%T Normal form of the wave group and inverse spectral theory
%J Journées équations aux dérivées partielles
%D 1998
%P 1-18
%I Université de Nantes
%U https://proceedings.centre-mersenne.org/item/JEDP_1998____A15_0/
%G en
%F JEDP_1998____A15_0
Steve Zelditch. Normal form of the wave group and inverse spectral theory. Journées équations aux dérivées partielles (1998), article  no. 15, 18 p. https://proceedings.centre-mersenne.org/item/JEDP_1998____A15_0/

[B.B] V.M.Babic, V.S. Buldyrev, Short-Wavelength Diffraction Theory, Springer Series on Wave Phenomena 4, Springer-Verlag, New York (1991) | MR | Zbl

[CV.1] Y. Colin De Verdière, 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 | Zbl

[CV.2] Y. Colin De Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent II. Le cas intégrable, Math.Zeit. 171 (1980), 51-73. | EuDML | MR | Zbl

[CV.3] Y. Colin De Verdière, Quasi-modes sur les variétés Riemanniennes, Inv. Math 43 (1977), 15-52. | EuDML | MR | Zbl

[D.G] J.J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. Math. 24 (1975), 39-80. | EuDML | MR | Zbl

[F.Z] G. Forni and S.Zelditch, Inverse spectral problem for surfaces of revolution II (in preparation).

[F.G] J.P. Francoise and V. Guillemin, On the period spectrum of a symplectic mapping, J. Fun. Anal. 100, (1991) 317-358. | MR | Zbl

[Go] C.Gordon, CBMS Lectures (1996).

[G.1] V. Guillemin, Wave trace invariants, Duke Math. J. 83 (1996), 287-352. | MR | Zbl

[G.2] V. Guillemin, Wave-trace invariants and a theorem of Zelditch, Duke Int.Math.Res.Not. 12 (1993), 303-308. | MR | Zbl

[G.M] V. Guillemin and R.B.Melrose, The Poisson summation formula for manifolds with boundary, Adv. in Math.32 (1979), 204-232. | MR | Zbl

[HoI-IV] L.Hörmander, Theory of Linear Partial Differential Operators I-IV, Springer-Verlag, New York (1985).

[Kac] M.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

[M.M] S.Marvizi and R.B.Melrose, Spectral invariants of convex planar regions, J. Diff.Geom. 17 (1982), 475-502. | MR | Zbl

[M] R.B. Melrose, The wave equation for a hypoelliptic operator with symplectic characteristics of codimension two, Journal D'Analyse Math. XLIV (1984/1985), 134-182. | MR | Zbl

[P] G.Popov, Length spectrum invariants of Riemannian manifolds, Math.Zeit. 213 (1993), 311-351. | MR | Zbl

[Sj] J.Sjöstrand, Semi-excited states in nondegenerate potential wells, Asym.An.6 (1992) 29-43. | MR | Zbl

[W] A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 (1977), 883-892. | MR | Zbl

[Z.1] S.Zelditch, Wave invariants at elliptic closed geodesics, Geom.Anal.Fun.Anal. 7 (1997), 145-213. | MR | Zbl

[Z.2] S.Zelditch, Wave invariants for non-degenerate closed geodesics, Geom.Anal.Fun.Anal. 8 (1998), 179-217. | MR | Zbl

[Z.3] S.Zelditch, Inverse spectral problem for surfaces of revolution (to appear in J.Diff.Geom.). | Zbl