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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|Summary:||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.|