Towards a Quantum Analog of Weak KAM Theory

We discuss a quantum analogue of Mathers minimization principle for Lagrangian dynamics, and provide some formal calculations suggesting the corresponding Euler–Lagrange equation. We then rigorously construct from the dual eigenfunctions of a certain non-selfadjoint operator a candidate for a minimizer, and recover aspects of weak KAM theory in the limit as h0. Regarding our state as a quasimode, we furthermore derive some error estimates, although it remains an open problem to improve these bounds.Supported in part by NSF Grant DMS-0070480 and by the Miller Institute for Basic Research in Science, UC Berkeley