# Geometry of Bicharacteristics

## Abstract

If *p* is of spectral type 1 on *Σ* there is no bicharacteristic with a limit point in *Σ*. When *p* is of spectral type 2 the spectral property of *F*_{ p } itself is not enough to determine completely the behavior of bicharacteristics and we need to look at the third order term of the Taylor expansion of *p* around the reference characteristic to obtain a complete picture of the behavior of bicharacteristics. We prove that there is no bicharacteristic with a limit point in *Σ* near *ρ* if and only if the cube of the vector field *H*_{ S }, introduced in Chap. 2, annihilates *p* on *Σ* near *ρ*. This suggests that the behavior of bicharacteristics near *Σ* closely relates to the possibility of microlocal factorization discussed in Chap. 2. In the last section, we prove that at every effectively hyperbolic characteristic there exists exactly two bicharacteristics that are transversal to *Σ* having which as a limit point. This makes the difference of the geometry of bicharacteristics clearer.

## References

- 3.E. Bernardi, A. Bove, Geometric results for a class of hyperbolic operators with double characteristics. Commun. Partial Differ. Equ.
**13**, 61–86 (1988)MathSciNetCrossRefMATHGoogle Scholar - 11.C. Briot, J.C. Bouquet, Recherches sur les propriétés des fonctions définies par des équations différentielles. J. Ec. Imp. Polytech.
**36**, 133–198 (1856)Google Scholar - 19.F.R. de Hoog, R. Weiss, On the boundary value problem for systems of ordinary differential equations with a singularity of the second kind. SIAM J. Math. Anal.
**11**, 41–60 (1980)MathSciNetCrossRefMATHGoogle Scholar - 33.L. Hörmander,
*The Analysis of Linear Partial Differential Operators. I*. Grundlehren Math. Wiss., vol. 274 (Springer, Berlin, 1983)Google Scholar - 40.V.Ja. Ivrii, The wellposed Cauchy problem for non-strictly hyperbolic operators, III. The energy integral. Trans. Moscow Math. Soc. (English transl.)
**34**, 149–168 (1978)Google Scholar - 48.N. Iwasaki, The Cauchy problem for effectively hyperbolic equations (remarks), in
*Hyperbolic Equations and Related Topics*(Academic, Boston, 1986), pp. 89–100CrossRefGoogle Scholar - 52.G. Komatsu, T. Nishitani, Continuation of bicharacteristics for effectively hyperbolic operators. Publ. Res. Inst. Math. Sci.
**28**, 885–911 (1992)MathSciNetCrossRefMATHGoogle Scholar - 72.T. Nishitani, Note on some non effectively hyperbolic operators. Sci. Rep. College Gen. Ed. Osaka Univ.
**32**(2), 9–17 (1983)MathSciNetMATHGoogle Scholar - 78.T. Nishitani, Non effectively hyperbolic operators, Hamilton map and bicharacteristics. J. Math. Kyoto Univ.
**44**, 55–98 (2004)CrossRefMATHGoogle Scholar - 87.T. Nishitani, A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators. Kyoto J. Math.
**55**, 281–297 (2015)MathSciNetCrossRefMATHGoogle Scholar