Generic properties of reflecting rays
Vesselin M. Petkov · 2016
This chapter establishes several properties of periodic reflecting rays in bounded domains and scattering rays in domains with bounded complements. These properties are used to investigate certain inverse spectral problems for generic domains. The chapter provides a general theorem that provides existence of residual sets of smooth embeddings of a given submanifold of Rn satisfying certain conditions. As a consequence of it, some elementary generic properties are derived for the two kinds of reflecting rays under consideration. In particular, the chapter shows that Herman Weyl's conjecture is true for generic bounded domains. It shows that for generic domains the reflecting rays under consideration are ordinary and non-degenerate. The lengths of any two distinct primitive periodic reflecting rays are independent over the rationals. For every x ∈ ∂Ω there exists at most one direction ξ ∈ Sn−1 such that (x, ξ) generates a periodic reflecting ray in Ω.