Paramétrix du problème mixte pour l'équation des ondes à l'intérieur d'un domaine convexe pour les bicaractéristiques

C∞spectral rigidity of the ellipse

“…Although the principal symbols are calculated in the interior in [Guillemin and Melrose 1979b, Proposition 5.1;Marvizi and Melrose 1982, Section 6;Petkov and Stoyanov 1992, Section 6], the results do not seem to be stated along the boundary (i.e., the symbols are not calculated at the boundary). The statements we need are contained in Theorem 3.1 of [Chazarain 1976] (and its proof), and we largely follow its presentation. We need to calculate the canonical relation and principal symbol of the wave group, its derivatives and their restrictions to the boundary.…”

“…For background we refer to [Guillemin and Melrose 1979b;Petkov and Stoyanov 1992;Chazarain 1973;1976;Hörmander 1985a, Theorem 23.1.4;1985b, Proposition 29.3.2]. For the application to spectral rigidity, we only need a microlocal description of wave kernels away from the glancing set, that is, in the hyperbolic set microlocally near periodic transversal reflecting rays.…”

“…For the application to spectral rigidity, we only need a microlocal description of wave kernels away from the glancing set, that is, in the hyperbolic set microlocally near periodic transversal reflecting rays. In these regions, there exists a microlocal parametrix due to Chazarain [1976], which is more fully analyzed in [Guillemin and Melrose 1979b;Petkov and Stoyanov 1992] and applied to the ellipse in [Guillemin and Melrose 1979a].…”

“…Notice we let j ∈ ‫ޚ‬ which is different from [Chazarain 1976] where j goes from 0 to ∞ and where the two graphs j ± and − j ± are combined. We know discuss these graphs more precisely.…”

“…We know discuss these graphs more precisely. We first recall some useful notation from [Chazarain 1976] with a slight adjustment. We have two Hamiltonian flows g ±t corresponding to the Hamiltonians ±|η|.…”