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Given a real semisimple connected Lie group $G$ and a discrete torsion-free subgroup $\Gamma < G$ we prove a precise connection between growth rates of the group $\Gamma$, polyhedral bounds on the joint spectrum of the ring of invariant differential operators, and the decay of matrix coefficients. In particular, this allows us to completely characterize temperedness of $L^2(\Gamma\backslash G)$ in this general setting., Comment: 22 pages, 3 figures, significantly sharpened v1

Decomposing complex unitary evolution into a series of constituent components is a cornerstone of practical quantum information processing. While the decompostion of an $n\times n$ unitary into a series of $2\times2$ subunitaries is well established (i.e. beamsplitters and phase shifters in linear optics), we show how this decomposition can be generalised into a series of $m\times m$ multimode devices, where $m>2$. If the cost associated with building each $m\times m$ multimode device is less than constructing with $\frac{m(m-1)}{2}$ individual $2\times 2$ devices, we show that the decomposition of large unitaries into $m\times m$ submatrices is is more resource efficient and exhibits a higher tolerance to errors, than its $2\times 2$ counterpart. This allows larger-scale unitaries to be constructed with lower errors, which is necessary for various tasks, not least Boson sampling, the quantum Fourier transform and quantum simulations.

We present a numerical algorithm for the computation of invariant Ruelle distributions on convex co-compact hyperbolic surfaces. This is achieved by exploiting the connection between invariant Ruelle distributions and residues of meromorphically continued weighted zeta functions established by the authors together with Barkhofen (2021). To make this applicable for numerics we express the weighted zeta as the logarithmic derivative of a suitable parameter dependent Fredholm determinant similar to Borthwick (2014). As an additional difficulty our transfer operator has to include a contracting direction which we account for with techniques developed by Rugh (1992). We achieve a further improvement in convergence speed for our algorithm in the case of surfaces with additional symmetries by proving and applying a symmetry reduction of weighted zeta functions., Comment: 52 pages, 25 figures

We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on vector spaces associated with the set of non backtracking paths in the graphs. The second kind of operators are averaging operators acting on vector spaces associated with the space of vertices of the graph. The choice of vector spaces reflects regularity properties. Our main results are correspondences between classical and quantum spectral objects as well as some automatic regularity properties for eigenfunctions of transfer operators., Comment: 37 pages, 2 figures

Let $X=X_1\times X_2$ be a product of two rank one symmetric spaces of non-compact type and $\Gamma$ a torsion-free discrete subgroup in $G_1\times G_2$. We show that the spectrum of $\Gamma \backslash X$ is related to the asymptotic growth of $\Gamma$ in the two direction defined by the two factors. We obtain that $L^2(\Gamma \backslash G)$ is tempered for large class of $\Gamma$., Comment: 17 pages, 1 figure

Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $\geq 1/4$. In this article we prove a generalization of this result for the joint $L^2$-eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces $\Gamma\backslash G/K$ of higher rank. We derive dynamical assumptions on the $\Gamma$-action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups., Comment: 15 pages, 5 figures, revised version with more explanations and figures

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., Comment: minor corrections

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., Comment: 25 pages, 4 figures; with an appendix by Sonja Barkhofen

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., Comment: Revised version with new appendix on vector bundles. 36 pages, 8 figures

We introduce a parameterized family of Poisson transforms on trees of bounded degree, construct explicit inverses for generic parameters, and characterize moderate growth of Laplace eigenfunctions by H\"older regularity of their boundary values., Comment: minor corrections of typos, to appear in the Journal of Spectral Theory

Given a general Anosov $\mathbb{R}^\kappa$ action on a closed manifold, we study properties of certain invariant measures that have recently been introduced in \cite{BGHW20} using the theory of Ruelle-Taylor resonances. We show that these measures share many properties of Sinai-Ruelle-Bowen measures for general Anosov flows such as smooth disintegrations along the unstable foliation, positive Lebesgue measure basins of attraction and a Bowen formula in terms of periodic orbits. Finally we show that if the action in the positive Weyl chamber is transitive, the measure is unique and has full support., Comment: Substancial revision after referees reports

For a compact Riemannian locally symmetric space $\Gamma\backslash G/K$ 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 $\Gamma$ 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 $G/K$., Comment: 23 pages, 6 figures

Let $G$ be a nilpotent, connected, simply connected Lie group with Lie algebra $\mathfrak g$, and $\pi$ a unitary representation of $G$. The goal is to prove that the wave front set of $\pi$ coincides with the asymptotic cone of the orbital support of $\pi$, i.e. $\mathrm{WF}(\pi)=\mathrm{AC}(\bigcup_{\sigma\in \mathrm{supp}(\pi)}\mathcal O_\sigma)$, where $\mathcal O_\sigma\subset i\mathfrak g^\ast$ is the coadjoint orbit associated to the irreducible unitary representation $\sigma\in \hat{G}$ by Kirillov., Comment: to appear in Journal of Functional Analysis

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., Comment: This is a shortened version of arXiv:1909.06183 but generalized to all constant curvature surfaces instead of negatively curved surfaces

Combining microlocal methods and a cohomological theory developped by J. Taylor, we define for $\mathbb{R}^\kappa$-Anosov actions a notion of joint Ruelle resonance spectrum. We prove that these Ruelle-Taylor resonances fit into a Fredholm theory, are intrinsic and form a discrete subset of $\mathbb{C}^\kappa$, with $\lambda=0$ being always a leading resonance. The joint resonant states at $0$ give rise to some new measures of SRB type and the mixing properties of these measures are related to the existence of purely imaginary resonances. The spectral theory developed in this article applies in particular to the case of Weyl chamber flows and provides a new way to study such flows., Comment: minor changes, to be published in Journal of the EMS

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 $SL_2(\mathbb{R})$., Comment: Minor corrections. The main result of the spectral convergence is also covered with a simplified proof and in a more general setting in arXiv:2011.06434. This version still contains explicit error estimates not contained in arXiv:2011.06434

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., Comment: 18 pages. To appear in Communications in Mathematical Physics

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., Comment: Minor changes, to appear at Annales Henri Lebesgue

We present new methods to construct a Ruelle-Pollicott spectrum for the geodesic flow on manifolds with strictly negative curvature and a finite number of hyperbolic cusps., Comment: detailed revision according to referee report

For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_\tau$ over the unit cosphere bundle $S^\ast\mathcal M$, we give a precise description of those classical (Pollicott-Ruelle) resonant states on $\mathbf V_\tau$ 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 $D(G,\sigma)$ on compatible associated vector bundles $\mathbf W_\sigma$ 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 $\tau$ and $\sigma$ defining the bundles $\mathbf V_\tau$ and $\mathbf W_\sigma$, 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 $\mathbf W_\sigma$. Our methods of proof are based on representation theory and Lie theory., Comment: minor corrections and updated references, to appear at IMRN

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., Comment: Minor corrections. Added a discussion of the special case of an elementary convex co-compact group. To appear in Mathematische Annalen

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 $\delta$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-\beta,0]$ is $O(R^{m(\beta,\delta)+})$, where the exponent $m(\beta,\delta)=\min(2\delta+2\beta+1-n,\delta)$ changes its behavior at $\beta=(n-1-\delta)/2$. In the case $\delta<(n-1)/2$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\mathrm{Im}\ \lambda\ > \delta-(n-1)/2\}$. 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., Comment: 42 pages, 10 figures; with an appendix by David Borthwick and Tobias Weich. Revised following suggestions of the referee. To appear in JEMS

We show that all generalized Pollicott-Ruelle resonant states of a topologically transitiv $C^\infty$-Anosov flow with an arbitrary $C^\infty$ potential, have full support., Comment: minor corrections/journal reference added, Ann. Henri Poincar\'e (2016)

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., Comment: 51 pages

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., Comment: Accepted to Advances in Mathematics

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., Comment: To appear in "Journal of Spectral Theory"

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.

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.

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., Comment: Updated references and some minor changes and corrections of typos

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 \nu on 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 \nu -> infinity. Some numerical computations with the truncated Gauss map illustrate these results., Comment: 65 pages