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Spectral Theory for Semiclassical Operators and Artificial Black Holes

  • Author(s): Hall, Michael Allen;
  • Advisor(s): Hitrik, Michael;
  • et al.

In this thesis we study several problems related to the spectral theory of semiclassical pseudodifferential operators, as well as artificial black holes in a curved spacetime. For non-selfadjoint perturbations of selfadjoint operators in dimension 2, we show that one can recover the (quantum) Birkhoff normal form of the operator near a Lagrangian torus satisfying a Diophantine condition from an appropriate portion of the spectrum, provided the unperturbed operator is known and under analyticity assumptions. Also working in dimension 2, we use a quantum version of the method of averaging, combined with techniques inspired by secular perturbation theory, to derive microlocal normal forms for selfadjoint semiclassical operators in dimension 2 with periodic classical flow. Finally, for stationary metrics in 2 space dimensions, we exhibit artificial black holes where the ergosphere and event horizon meet at isolated points, and which display a complicated dynamical structure.}

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