Locally symmetric spaces and transfer operators
Final Report Abstract
This project was dedicated to the development of transfer operator techniques for the study of mathematical quantum chaos on Riemannian locally symmetric spaces. A first focus was on the investigation of the interaction between the geometric-dynamical entities and the spectral entities of hyperbolic surfaces. A second focus was on carrying out crucial first steps towards a generalization to more general spaces, in particular to locally symmetric spaces of higher rank. We achieved the following results, among others. For a certain family of hyperbolic surfaces of infinite area, namely the Hecke triangle surfaces of infinite area, we developed explicit cohomological interpretations of automorphic functions with different behavior at the ends of the spaces. We further constructed isomorphisms between the cohomology spaces and certain precise spaces of eigenfunctions of the slow transfer operators for the Hecke triangle groups. Moreover, we established explicit isomorphisms between spaces of eigenfunctions of the slow transfer operators and eigenfunctions of the fast transfer operators for all Hecke triangle groups. These results contribute to the understanding of the classical-quantum correspondence between the geodesic flow on the one hand and Laplace eigenfunctions, resonant states and resonances on the other hand. All constructions and proofs in these results are conceptual and therefore can be generalized to other Fuchsian groups without too much additional work (which we left to future considerations). A long-standing conjecture (Sjöstrand, Lu–Sridhar–Zworski) predicts the existence of a certain fractal Weyl law for the counting function of the resonances of hyperbolic surfaces. We provided the first instance of an upper fractal Weyl bound for a certain family of infinite-area hyperbolic surfaces with cusps. The proof combines methods from thermodynamic formalism and analysis, and takes advantage of the fast transfer operators developed previously by the grant holder. In addition, we discussed, using similar methods, the behavior of resonance counting functions for Schottky surfaces under transition to covers. Generalized modular forms are requested by applications in mathematics and physics. We showed the convergence (on right half spaces) of Selberg zeta functions that are twisted by representations of non-expanding cusp monodromy. Further, under the condition of the existence of a transfer operator representation of the Selberg zeta function, we established its meromorphic continuation to the whole complex plane. It is expected that the singularities of these zeta functions are closely related to spectral parameters of generalized modular forms or twisted automorphic functions. We provided the first steps towards such an interpretation. We extended significantly the realm of the dual slow/fast transfer operator approaches, which are at the basis of the previously mentioned results. The passage from slow to fast transfer operators, even though very technical, is highly conceptual. A further generalization to higher-dimensional spaces will be possible. Furthermore, we developed an algorithm for the computation of resonances of Schottky surfaces that is based on Lagrange–Chebyshev approximation and transfer operator techniques. It pushes further the frontier of numerical investigations of these resonances, and allowed us to make some new (experimental) observations. Finally, we provided first examples of discretizations, symbolic dynamics and transfer operators for the Weyl chamber flow on certain families of locally symmetric spaces. These constructions follow certain specific concepts and can easily be extended to more general spaces. They are the first crucial results towards a classical-quantum correspondence for spaces of higher rank, whose investigations will be continued now.
Publications
- Zero is a resonance of every Schottky surface
Anke D. Pohl, A. Adam and A. Weiße
- Dynamics of geodesics, and Maass cusp forms (with D. Zagier), Enseign. Math.
Anke D. Pohl, D. Zagier
- Eigenfunctions of transfer operators and automorphic forms for Hecke triangle groups of infinite covolume (with R. Bruggeman), AMS Memoirs
Anke D. Pohl
- Fractal Weyl bounds and Hecke triangle groups, Electron. Res. Announc. Math. Sci. 26, 24–35 (2019)
Anke D. Pohl, F. Naud and L. Soares
(See online at https://doi.org/10.3934/era.2019.26.003) - A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta
functions
Anke D. Pohl,A. Adam
- A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions, Ergodic Theory Dynam. Systems 40 (2020), no. 3, 612–662
Anke D. Pohl, A. Adam
(See online at https://doi.org/10.1017/etds.2018.51) - Density of resonances for covers of Schottky surfaces, J. Spectr. Theory 10 (2020), no. 3, 1053–1101
Anke D. Pohl, L. Soares
(See online at https://doi.org/10.4171/JST/321) - Eisenstein series twisted with non-expanding cusp monodromies, Ramanujan J. 51 (2020), no. 3, 649–670
Anke D. Pohl, K. Fedosova
(See online at https://doi.org/10.1007/s11139-019-00205-5) - Numerical resonances for Schottky surfaces via Lagrange–Chebyshev approximation, Stoch. Dyn.
Anke D. Pohl, O. Bandtlow, T. Schick, A. Weiße
(See online at https://doi.org/10.1142/S0219493721400050) - Symbolic dynamics and transfer operators for Weyl chamber flows: a class of examples
Anke D. Pohl
