Singular perturbation of symbolic flows and poles of the zeta functions.

*(English)*Zbl 0708.58019This 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.

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

##### MSC:

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 |