Higher rank quantum-classical correspondence - Hilgert, Joachim; Weich, Tobias; Wolf, Lasse Lennart (Analysis & PDE)
For a compact Riemannian locally symmetric space $Γ\backslash G/K$ of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle-Taylor resonances and establish a spectral gap which is uniform in $Γ$ if $G/K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G/K$.

Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems - Schütte, Philipp; Weich, Tobias; Barkhofen, Sonja (Communications in Mathematical Physics)
In this article we prove meromorphic continuation of weighted zeta functions in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (2016). We obtain a residue formula proving equality between residues of weighted zetas and invariant Ruelle distributions. We combine this equality with results of Guillarmou, Hilgert and Weich (2021) in order to relate the residues to Patterson-Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems.

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds - Cekić, Mihajlo; Delarue, Benjamin; Dyatlov, Semyon; Paternain, Gabriel P. (Inventiones mathematicae)
Abstract We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $$Σ $$ Σ with Betti number $$b_1$$ b 1 , the order of vanishing of the Ruelle zeta function at zero equals $$4-b_1$$ 4 - b 1 , while in the hyperbolic case it is equal to $$4-2b_1$$ 4 - 2 b 1 . This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $$SΣ $$ S Σ with harmonic 1-forms on $$Σ $$ Σ .
Keywords: General Mathematics

Ruelle–Pollicott resonances for manifolds with hyperbolic cusps - Guedes Bonthonneau, Yannick; Weich, Tobias (Journal of the European Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

Semiclassical formulae For Wigner distributions - Barkhofen, Sonja; Schütte, Philipp; Weich, Tobias (Journal of Physics A: Mathematical and Theoretical)
In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems.

Poisson transforms for trees of bounded degree - Bux, Kai-Uwe; Hilgert, Joachim; Weich, Tobias (Journal of Spectral Theory)
Keywords: Geometry and Topology; Mathematical Physics; Statistical and Nonlinear Physics

Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces - Olbrich, Martin; Palmirotta, Guendalina (Annals of Global Analysis and Geometry)

A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on $G/K$ - Olbrich, Martin; Palmirotta, Guendalina (Journal of Lie theory)
We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type $X = G/K$. We prove a characterisation of their range. In fact, from Delorme's Paley-Wiener theorem for compactly supported smooth functions on a real reductive group of Harish-Chandra class, we deduce topological Paley-Wiener and Paley-Wiener-Schwartz theorems for sections.

Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions - Delarue, Benjamin; Ramacher, Pablo (Journal of Symplectic Geometry)

Resonances and weighted zeta functions for obstacle scattering via smooth models - Schütte, Philipp; Weich, Tobias; Delarue, Benjamin
We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces - Küster, Benjamin; Weich, Tobias (International Mathematics Research Notices)
Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{τ }$ over the unit cosphere bundle $S^{* }\mathcal M$, 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 $S^{* }\mathcal M$. 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.
Keywords: General Mathematics

High frequency limits for invariant Ruelle densities - Guillarmou, Colin; Hilgert, Joachim; Weich, Tobias (Annales Henri Lebesgue)

Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds - Guillarmou, Colin; Küster, Benjamin (Annales Henri Poincaré)
Keywords: Mathematical Physics; Nuclear and High Energy Physics; Statistical and Nonlinear Physics

Pollicott-Ruelle Resonant States and Betti Numbers - Küster, Benjamin; Weich, Tobias (Communications in Mathematical Physics)
Abstract Given a closed orientable hyperbolic manifold of dimension $$\ne 3$$ ≠ 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.
Keywords: Mathematical Physics; Statistical and Nonlinear Physics

Pollicott-Ruelle Resonant States and Betti Numbers - Küster, Benjamin; Weich, Tobias (Communications in Mathematical Physics)
Abstract Given a closed orientable hyperbolic manifold of dimension $$\ne 3$$ ≠ 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.
Keywords: Mathematical Physics; Statistical and Nonlinear Physics

Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich - Dyatlov, Semyon; Borthwick, David; Weich, Tobias (Journal of the European Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces - Kolb, Martin; Weich, Tobias; Wolf, Lasse Lennart (arXiv:1909.06183)
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 constant negatively 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. In addition, we give explicit error estimates for the convergence to equilibrium. The proofs are based on noncommutative harmonic analysis of $SL_2(\mathbb{R})$.

Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces - Schütte, Philipp

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces - Küster, Benjamin; Weich, Tobias (International Mathematics Research Notices)
Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{τ }$ over the unit cosphere bundle $S^{* }\mathcal M$, 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 $S^{* }\mathcal M$. 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.
Keywords: General Mathematics

Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks - Schütte, Philipp