zbMATH — the first resource for mathematics

Singular perturbation of symbolic flows and poles of the zeta functions. (English) Zbl 0708.58019
This work is motivated by studies of scattering by an obstacle \({\mathcal O}\) which consists of several strictly convex bodies. In this context, a function \(F_ D(s)\) can be defined by means of the geometry of periodic rays in the exterior of \({\mathcal O}\). A theorem of Ikawa says that if \(F_ D(s)\) cannot be analytically continued to an entire function, then the scattering matrix S(z) for \({\mathcal O}\) has an infinite number of poles in the half-plane \(\{\) z: \(Im(z)<\alpha \}\) for some \(\alpha >0\). This, in turn, means that the modified Lax-Phillips conjecture is valid for \({\mathcal O}.\)
In general, it is difficult to show that an analytic continuation of \(F_ D(s)\) to an entire function is impossible. However, \(F_ D(s)\) is closely related to the zeta function \(\zeta\) (s) of a certain associated symbolic flow; specifically, the singularities of -(d/ds)log \(\zeta\) (s) in a specified domain coincide with those of \(F_ D(s)\). Thus, consideration of the singularities of \(F_ D(s)\) is essentially reduced to analysis of \(\zeta\) (s). Unfortunately, it is also difficult to find the singularities of -(d/ds)log \(\zeta\) (s).
The author shows that, if the convex bodies are small compared to the distances between them, then it is possible to find a pole of \(\zeta\) (s) corresponding to \({\mathcal O}\). The author’s main theorem is used to show that the modified Lax-Phillips conjecture is valid for obstacles of this type.
Reviewer: W.J.Satzer jun

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C10 Dynamics induced by flows and semiflows
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
35P25 Scattering theory for PDEs