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Joachim Hilgert

Publications and source records attributed to Joachim Hilgert.

At least 19 recordsLinked to original sources
arXiv2026

Spectral theory for transfer operators on compact quotients of Euclidean buildings

In this paper we generalize the geodesic flow on (finite) homogeneous graphs to a multiparameter flow on compact quotients of Euclidean buildings. Then we study the joint spectra of the associated transfer operators acting on suitable Lipschitz spaces. The main result says that outside an arbitrarily small neighborhood of zero in the set of spectral parameters the Taylor spectrum of the commuting family of transfer operators is contained in the joint point spectrum.

math.DS↗
arXiv2026

Fourier-Helgason transform as infinite geodesic time limit in geometric quantization

The Fourier-Helgason (FH) transform for a noncompact symmetric space $G/K$ establishes the direct integral decomposition of the unitary representation of $G$ on $L^2(G/K)$ into irreducible principal series representations. By applying techniques of geometric quantization to the symplectic manifold $T^*(G/K),$ Lisiecki in 1987 gave a geometric interpretation of the FH transform in the case when $G$ is complex. He defined for general $G$ a ''horizontal'' polarization on $T^*(G/K)$ and showed that, for complex $G$, the Blattner-Kostant-Sternberg (BKS) pairing between the Schrödinger vertical polarization Hilbert space, $L^2(G/K)$, and the Hilbert space of horizontally polarized functions coincides with the FH transform. However, in the same paper, Lisiecki showed that for noncomplex Lie groups the BKS pairing is nonequivalent to the FH transform and nonunitary in general. In the present paper, we resolve this discrepancy between the FH transform and geometric quantization in the case when $G$ is not complex. First, we show that the horizontal polarization is the infinite-time limit of the push-forward of the vertical polarization with respect to the geodesic flow for a $G$-invariant Riemannian metric. Then we lift the geodesic flow to an intertwining unitary parallel transport on the quantum bundle that we call quantum geodesic transform (QGT). Finally we show that the QGT has a well-defined limit, as the geodesic time goes to infinity, and that it is equal, up to the phase of the Harish-Chandra $c$-function and an irrelevant multiplicative constant, to the FH transform.

math.SG↗
arXiv2024

Quantum resonances and scattering poles of classical rank one locally symmetric spaces

For negatively curved symmetric spaces it is known from [Hansen-Hilgert-Parthasarathy,2019] that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-cocompact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.

math.SP↗
arXiv2024

Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

In this paper, we describe holomorphic quantizations of the cotangent bundle of a symmetric space of compact type $T^*(U/K)\cong U_\mathbb{C}/K_\mathbb{C}$, along Mabuchi rays of $U$-invariant Kähler structures. At infinite geodesic time, the Kähler polarizations converge to a mixed polarization $\mathcal{P}_\infty$. We show how a generalized coherent state transform relates the quantizations along the Mabuchi geodesics such that holomorphic sections converge, as geodesic time goes to infinity, to distributional $\mathcal{P}_\infty$-polarized sections. Unlike in the case of $T^*U$, the gCST mapping from the Hilbert space of vertically polarized sections are not asymptotically unitary due to the appearance of representation dependent factors associated to the isotypical decomposition for the $U$-action. In agreement with the general program outlined in [Bai+23], we also describe how the quantization in the limit polarization $\mathcal{P}_\infty$ is given by the direct sum of the quantizations for all the symplectic reductions relative to the invariant torus action associated to the Hamiltonian action of $U$.

math.SG↗
arXiv2024

Quantum-Classical Correspondences for Locally Symmetric Spaces

We review the status of a program, outlined and motivated in the introduction, for the study of correspondences between spectral invariants of partially hyperbolic flows on locally symmetric spaces and their quantizations. Further we formulate a number of concrete problems which may be viewed as possible further steps to be taken in order to complete the program.

math.DS↗
arXiv2023

A Pairing Formula for Resonant States on Finite Regular Graphs

On a finite regular graph, (co)resonant states are eigendistributions of the transfer operator associated to the shift on one-sided infinite non-backtracking paths. We introduce two pairings of resonant and coresonant states, the vertex pairing which involves only the dependence on the initial/terminal vertex of the path, and the geodesic pairing which is given by integrating over all geodesics the evaluation of the coresonant state on the first half of the geodesic times the resonant state on the second half. The main result is that these two pairings coincide up to a constant which depends on the resonance, i.e. the corresponding eigenvalue of the transfer operator.

math.SP↗
arXiv2023

Edge Laplacians and Edge Poisson Transforms for Graphs

For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on one-sided infinite non-backtracking paths. Interpreting the transfer operator as a classical dynamical system and the Laplace operator as its quantization, this result can be viewed as a quantum-classical correspondence. In contrast to previously established quantum-classical correspondences for the vertex Laplacian which exclude certain exceptional spectral parameters, our correspondence is valid for all parameters. This allows us to relate certain spectral quantities to topological properties of the graph such as the cyclomatic number and the 2-colorability. The quantum-classical correspondence for the edge Laplacian is induced by an edge Poisson transform on the universal covering of the graph which is a tree of bounded degree. In the special case of regular trees, we relate both the vertex and the edge Poisson transform to the representation theory of the automorphism group of the tree and study associated operator valued Hecke algebras.

math.SP↗
arXiv2023

Spectral correspondences for finite graphs without dead ends

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.

math.SP↗
arXiv2023

Ruelle-Taylor resonances of Anosov actions

Combining microlocal methods and a cohomological theory developped by J. Taylor, we define for $\mathbb{R}^κ$-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 $\mathbb{C}^κ$, with $λ=0$ 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.

math.DS↗
arXiv2023

Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter--Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact semi-simple groups $K$. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of $K\times K$-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action. Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for $K$ with the Borel--Weil description of irreducible representations of $K$.

math.SG↗
arXiv2022

Aspects of control theory on infinite-dimensional Lie groups and G-manifolds

We develop aspects of geometric control theory on Lie groups G which may be infinite dimensional, and on smooth G-manifolds M modelled on locally convex spaces. As a tool, we discuss existence and uniqueness questions for differential equations on M given by time-dependent fundamental vector fields which are L^1 in time. We then discuss the closures of reachable sets in M for controls in the Lie algebra of G, or within a compact convex subset of the Lie algebra. Regularity properties of the Lie group G play an important role.

math.FA↗
arXiv2022

Poisson Transforms for Trees of Bounded Degree

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 Hölder regularity of their boundary values.

math.SP↗
arXiv2021

Spectral Correspondences for Rank One Locally Symmetric Spaces -- The Case of Exceptional Parameters

In this paper we complete the program of relating the Laplace spectrum for rank one compact locally symmetric spaces with the first band Ruelle-Pollicott resonances of the geodesic flow on its sphere bundle. This program was started by Flaminio and Forni for hyperbolic surfaces, continued by Dyatlov, Faure and Guillarmou for real hyperbolic spaces and by Guillarmou, Hilgert and Weich for general rank one spaces. Except for the case of hyperbolic surfaces a countable set of exceptional spectral parameters always left untreated since the corresponding Poisson transforms are neither injective nor surjective. We use vector valued Poisson transforms to treat also the exceptional spectral parameters. For surfaces the exceptional spectral parameters lead to discrete series representations of $\mathrm{SL}(2,\mathbb R)$. In higher dimensions the situation is more complicated, but can be described completely.

math.SP↗
arXiv2021

Higher rank quantum-classical correspondence

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 $G/K$.

math.DS↗
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↗
arXiv2019

Strongly symmetric spectral convex bodies are Jordan algebra state spaces

We show that the strongly symmetric spectral convex compact sets are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. Spectrality is the property that every state has a convex decomposition into perfectly distinguishable pure states; strong symmetry is transitivity, for each integer N, of the affine automorphism group of the state space on lists of N perfectly distinguishable pure states. Additional assumptions combine with this theorem to give simple characterizations of finite-dimensional complex quantum state space. Important aspects of quantum and classical thermodynamics and of query complexity have been generalized to classes of general probabilistic theories (GPTs) satisfying natural postulates including or implying spectrality and strong symmetry; our result shows that these apply to a narrower class of theories than might have been hoped. Sorkin's notion of irreducibly k-th order interference has been studied in the GPT framework and looked for in experiments. Our result shows that the assumption of no higher-order (k > 2) interference, used along with spectrality and strong symmetry to characterize the same class of Jordan-algebraic convex sets by Barnum, Mueller, and Ududec in arXiv:1403.4147, was superfluous. It also implies that Lee and Selby's extension, on the assumption that interference has fixed maximal degree k, of the important order square root of N lower bound on the quantum black-box query complexity of searching N possibilities for one having a desired property (which is achieved by Grover's quantum algorithm), to a class of theories satisfying certain postulates allowing the formulation of a generalized notion of query algorithm, actually applies in the Jordan-algebraic setting where higher-order interference is not possible.

math-ph↗
arXiv2018

Boundary Values of Eigenfunctions on Riemannian Symmetric Spaces

We give a new and self-contained proof of a generic version of the (former) Helgason conjecture. It says that for generic spectral parameters the Poisson transform is a topological isomorphism, with the inverse given by a boundary value map. Following Oshima's approach to a simplified definition of boundary values, and using the earlier work of Baouendi and Goulaouic on Fuchsian type equations, our proof is along the lines of our earlier work in the rank one distributional case, and works for both the hyperfunction and the distribution setting.

math.RT↗