Singular Equivariant Spectral Asymptotics of Schrödinger Operators in R^n and Resonances of Schottky Surfaces
This work consists of four self-containedly presented parts. In the first part we prove equivariant spectral asymptotics for h-pseudo- differential operators for compact orthogonal group actions generalizing re- sults of El-Houakmi and Helffer (1991) and Cassanas (2006). Using recent results for...
Reine und Angewandte Mathematik
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|Zusammenfassung:||This work consists of four self-containedly presented parts. In the first part we prove equivariant spectral asymptotics for h-pseudo- differential operators for compact orthogonal group actions generalizing re- sults of El-Houakmi and Helffer (1991) and Cassanas (2006). Using recent results for certain oscillatory integrals with singular critical sets (Ramacher 2010) we can deduce a weak equivariant Weyl law. Furthermore, we can prove a complete asymptotic expansion for the Gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the Hamilton flow. In the second and third part we study resonance chains which have been observed in many different physical and mathematical scattering problems. In the second part we present a mathematical rigorous study of the reso- nance chains on three funneled Schottky surfaces. We prove the analyticity of the generalized zeta function which provide the central mathematical tool for understanding the resonance chains. Furthermore we prove for a fixed ratio between the funnel lengths and in the limit of large lengths that after a suitable rescaling the resonances in a bounded domain align equidistantly along certain lines. The position of these lines is given by the zeros of an explicit polynomial which only depends on the ratio of the funnel lengths. In the third part we provide a unifying approach to these resonance chains by generalizing dynamical zeta functions. By means of a detailed numerical study we show that these generalized zeta functions explain the mechanism that creates the chains of quantum resonance and classical Ruelle resonances for 3-disk systems as well as geometric resonances on Schottky surfaces. We also present a direct system-intrinsic definition of the continuous lines on which the resonances are strung together as a projection of an analytic vari- ety. Additionally, this approach shows that the existence of resonance chains is directly related to a clustering of the classical length spectrum on multiples of a base length. Finally, this link is used to construct new examples where several different structures of resonance chains coexist. The fourth part deals with a symmetry factorization of dynamical zeta functions for holomorphic iterated function schemes. We introduce the no- tion of a finite symmetry group for these iterated function schemes and prove that the dynamical zeta function factorizes into entire symmetry reduced zeta functions that are parametrized by the irreducible characters of the symme- try group. Under an assumption on the group action on the symbols of the symbolic dynamics, we are able to simplify the formulas for the symmetry reduced zeta functions considerably. As an application we apply the symme- try factorization to Selberg zeta functions of symmetric n-funneled Schottky surfaces and show that the symmetry reduction simplifies the numerical cal- culation of the resonances strongly.|