Towards a Quantum Analog of Weak KAM Theory

We discuss a quantum analogue of Mather’s 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 ψ for a minimizer, and recover aspects of ‘‘weak KAM’’ theory in the limit as h→0. Regarding our state ψ as a quasimode, we furthermore derive some error estimates, although it remains an open problem to improve these bounds.