In ray approximations of boundary integral kernels, unitary transfer operators arise. The existence of transfer operators implies trace formulas for the fluctuating part of the spectral density and due to unitarity a reduction of the number of periodic orbits used in these formulas. We introduce the transfer operator method and discuss its applications in elasticity in the case of plate vibrations.

Introduction

In the construction of fluctuating spectral densities (Gutzwiller 1990, Brack & Bhaduri 1997, Stöckmann 1999), the physicist Gutzwiller took a trace of a unitary evolution operator, the Green's function in path integral form. The resulting trace formulas are expressed as a sum over periodic orbits. Later, in the setting of resonators it was discovered that the more conventional boundary integral kernels (Kitahara 1985, Bonnet 1995) could do the same job and that these kernels could also be made unitary.

One such approach is the transfer matrix of Bogomolny (Bogomolny 1992, Boasman 1994). Such ray kernels contain the essence of semiclassical approximations and can be used for numerics even without periodic orbits. Furthermore, these operators are unitary, leading to a truncation in the trace formula (called resurgence) of the infinitely many periodic orbits of the system in question (Berry & Keating 1990, Georgeot & Prange 1995). Another unitary approach is inside–outside duality based on scattering (Smilansky 1995, Smilansky & Ussishkin 1996).