Transfer operators via semiclassical methods

• Applicant: Professor Dr. Joachim Hilgert
• Subject Area: Mathematics
• Term: from 2014 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 252019602

• Final Report Year: 2017

Final Report Abstract

The research project focused mainly on questions in pure mathematics, that are strongly motivated by the following physical questions: Given a physical system, like for example a microdisk laser, what are the relation between the classical and quantum mechanical behavior? Transferring this correspondence to the mathematical world, the study of the classical behavior is a question in the field of dynamical system theory, whereas the quantum mechanical description leads to questions in spectral theory. Transferring this classical-quantum correspondence to the mathematical world thus means to establish a rigorous link between objects in different mathematical sub-disciplines such as spectral theory and dynamical systems. A main result was to establish such a rigorous classical-quantum correspondence on locally symmetric spaces. These spaces provide an excellent framework as they allow to apply powerful mathematical tools from harmonic analysis and representation theory. This aspect brought a third mathematical sub-discipline into the project and emphasized its interdisciplinary character. One of the main achievements of the project was to establish rigorous and exact relation of so called classical resonances and quantum resonance on these locally symmetric spaces. While for a real physical system, such a correspondence cannot be expected to be exactly true, one can hope that these explicit newly established correspondences still hold approximately in certain realistic physical systems. These could in a long term strongly improve our understanding of the physical correspondence between classical and quantum mechanics.

Publications

• Asymptotic spectral gap for open partially expanding maps
  F. Faure and T. Weich
  
• Classical and quantum resonances on hyperbolic surfaces
  C. Guillarmou, J. Hilgert and T. Weich
  (See online at https://doi.org/10.1007/s00208-017-1576-5)
• Resonance chains and geometric limits on Schottky surfaces. Communications in Mathematical Physics: Volume 337, Issue 2 (2015), Page 727-765
  T. Weich
  (See online at https://doi.org/10.1007/s00220-015-2359-z)
• Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions. Journal of Spectral Theory 6(2) p. 267-329
  D. Borthwick and T. Weich
  (See online at https://dx.doi.org/10.4171/JST/125)
• On the support of Pollicott-Ruelle resonant states for Anosov flows. Annales Henri Poincaré 18 (2017), 37-52
  T. Weich
  
• Wave Front Sets of Reductive Lie Group Representations III. Advances of Mathematics 313 (2017) 176-236
  B. Harris and T. Weich
  (See online at https://doi.org/10.1016/j.aim.2017.03.025)