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 natural pairings of resonant and coresonant states and prove that they coincide up to a constant which depends on the resonance. Published in Math. Ann., 2025.
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. Published in Potential Anal., 2025.
In this paper we realize all irreducible unitary representations of the group \(\mathrm{SO}_0(n+1,1)\) on explicit Hilbert spaces of vector-valued \(L^2\)-functions on \(\mathbb{R}^n\setminus\{0\}\). Preprint, 2024.
Unitary transformations are the basis of quantum information processing and quantum simulation. Typically these large unitaries are constructed from a decomposition into a collection of smaller \(2\times 2\) unitaries. In this paper we show how for any \(m>2\) this decomposition can be generalized to \(m\times m\) subunitaries. Published in Phys. Rev. Research, 2024.
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. Published in JEP, 2023.