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Tobias Weich

Publications and source records attributed to Tobias Weich.

33 records · Page 2Linked to original sources
arXiv2020

Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature

The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $M$ is a stochastic process that models a random perturbation of the geodesic flow. If $M$ is a orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $L^2$-spectrum of the infinitesimal generator of a time rescaled version of the process converges to the Laplace spectrum of the base manifold.

math.SP↗
arXiv2020

High frequency limits for invariant Ruelle densities

We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank one. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances which also leads to a new description of Patterson-Sullivan distributions.

math.AP↗
arXiv2020

Pollicott-Ruelle resonant states and Betti numbers

Given a closed orientable hyperbolic manifold of dimension $\neq 3$ we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numbers. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers.

math.SP↗
arXiv2019

Quantum-classical correspondence on associated vector bundles over locally symmetric spaces

For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_τ$ over the unit cosphere bundle $S^\ast\mathcal M$, we give a precise description of those classical (Pollicott-Ruelle) resonant states on $\mathbf V_τ$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^\ast\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,σ)$ on compatible associated vector bundles $\mathbf W_σ$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott-Ruelle spectrum. Further, under some mild assumptions on the representations $τ$ and $σ$ defining the bundles $\mathbf V_τ$ and $\mathbf W_σ$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott-Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_σ$. Our methods of proof are based on representation theory and Lie theory.

math.SP↗
arXiv2017

Asymptotic spectral gap for open partially expanding maps

We consider a $\mathbb{R}$-extension of one dimensional uniformly expanding open dynamical systems and prove a new explicit estimate for the asymptotic spectral gap. To get these results, we use a new application of a "global normal form" for the dynamical system, a "semiclassical expression beyond the Ehrenfest time" that expresses the transfer operator at large time as a sum over rank one operators (each is associated to one orbit). In this paper we establish the validity of the so-called "diagonal approximation" up to twice the local Ehrenfest time.

math.DS↗
arXiv2017

Classical and quantum resonances for hyperbolic surfaces

For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.

math.DS↗
arXiv2017

Improved fractal Weyl bounds for hyperbolic manifolds

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $δ$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-β,0]$ is $O(R^{m(β,δ)+})$, where the exponent $m(β,δ)=\min(2δ+2β+1-n,δ)$ changes its behavior at $β=(n-1-δ)/2$. In the case $δ<(n-1)/2$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\mathrm{Im}\ λ > δ-(n-1)/2\}$. Both results use the fractal uncertainty principle point of view recently introduced in [arXiv:1504.06589]. The appendix presents numerical evidence for the Weyl upper bound.

math.SP↗
arXiv2017

Wave Front Sets of Reductive Lie Group Representations III

Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of $L^2(X)$. Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on $X$. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.

math.RT↗
arXiv2016

Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions

Given a holomorphic iterated function scheme with a finite symmetry group $G$, we show that the associated dynamical zeta function factorizes into symmetry-reduced analytic zeta functions that are parametrized by the unitary irreducible representations of $G$. We show that this factorization implies a factorization of the Selberg zeta function on symmetric $n$-funneled surfaces and that the symmetry factorization simplifies the numerical calculations of the resonances by several orders of magnitude. As an application this allows us to provide a detailed study of the spectral gap and we observe for the first time the existence of a macroscopic spectral gap on Schottky surfaces.

math.SP↗
arXiv2014

Equivariant spectral asymptotics for h-pseudodifferential operators

We prove equivariant spectral asymptotics for $ h$-pseudodifferential operators for compact orthogonal group actions generalizing results 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.

math-ph↗
arXiv2014

Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum

In many non-integrable open systems in physics and mathematics resonances have been found to be surprisingly ordered along curved lines in the complex plane. In this article 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 variety. 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.

math-ph↗
arXiv2014

Resonance chains and geometric limits on Schottky surfaces

Resonance chains have been observed in many different physical and mathematical scattering problems. Recently numerical studies linked the phenomenon of resonances chains to an approximate clustering of the length spectrum on integer multiples of a base length. A canonical example of such a scattering system is provided by 3-funneled hyperbolic surfaces where the lengths of the three geodesics around the funnels have rational ratios. In this article we present a mathematical rigorous study of the resonance chains for these systems. 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 fun- nel 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.

math-ph↗
arXiv2013

Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

We consider a simple model of an open partially expanding map. Its trapped set K in phase space is a fractal set. We first show that there is a well defined discrete spectrum of Ruelle resonances which describes the asymptotics of correlation functions for large time and which is parametrized by the Fourier component νon the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call "minimal captivity". This hypothesis is stable under perturbations and means that the dynamics is univalued on a neighborhood of K. Under this hypothesis we show the existence of an asymptotic spectral gap and a Fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit ν-> infinity. Some numerical computations with the truncated Gauss map illustrate these results.

math-ph↗