Exact Results in Quantum Theory

Certain well-known techniques in quantum mechanics fail when one considers non-self-adjoint Hamiltonians (which appear, for instance, in kinetic theory). Elementary models include the Davies operator / complex harmonic oscillator -(d/dx)^2 + i x^2 and the harmonic oscillator with complex shift -(d/dx)^2 + x^2 + ix.In particular, the decomposition in eigenfunctions generally diverges and the Schrödinger evolution is no longer mass-preserving. We will discuss how complex extension of wave-packet decompositions gives us sharp estimates controlling these phenomena. In particular, we will discuss a recent result (joint with B. Mityagin and P. Siegl) on the hypoelliptic Laplacian on the circle, drawing from other works (with A. Aleman, M. Hitrik, K. Pravda-Starov, and J. Sjöstrand) and fundamental classical works by J. Sjöstrand and L. Hörmander.