Scattering Theory for Leaky Tori

Abstract

The leaky tori model for scattering theory in mesoscopic systems was introduced by Gutzwiller in 1983. Müller’s class of admissable surfaces subsumes the leaky tori models of Gutzwiller. In addition, they form a natural generalization of locally symmetric spaces of constant negative curvature and finite volume. In this chapter we review Müller’s spaces in Section 1. In Section 2 scattering operators are developed. In Section 3 Weyl’s law for mesoscopic systems is discussed. In Section 4 Miiller’s trace formula is presented. In Sections 5 to 8 work of Guillopé and Zworski (1994) on scattering theory on hyperbolic half-cylinders is developed. Their work uses some classical results of Pöschl and Teller which allows one to explicitly deduce the resonance set for the Laplacian on the hyperbolic half-cylinder. In Section 9 we briefly review related results on scattering theory for two strictly convex bodies. And in Section 10 we return to the Gutzwiller trace formula and look at including diffraction terms.