Convergence of a quantum normal form and an exact quantization formula

TL;DR

This paper proves the uniform convergence of a quantum normal form for a perturbed Schrödinger operator with Diophantine frequencies, leading to an exact quantization formula and a convergence criterion for the Birkhoff normal form.

Contribution

It establishes the convergence of the quantum normal form and derives an exact spectrum quantization formula for a class of holomorphic perturbations.

Findings

Quantum normal form converges uniformly in Planck's constant.

Exact quantization formula for the quantum spectrum is obtained.

Provides a convergence criterion for the Birkhoff normal form.

Abstract

We consider the Schr\"odinger operator defined by the quantization of the linear flow of diophantine frequencies over the l-dimensional torus, perturbed by a holomorphic potential which depends on the actions only through their particular linear combination defining the Hamiltonian of the linear flow. We prove that the corresponding quantum normal form converges uniformly with respect to the Planck constant. This result simultaneously yields an exact quantization formula for the quantum spectrum, as well as a convergence criterion for the Birkhoff normal form, valid for a class of perturbations holomorphic away from the origin.