Semiclassical Birkhoff-Gustavson normal forms and spectral asymptotics for nearly resonant Schr\"odinger operators

TL;DR

This paper introduces a method combining near resonance analysis and Birkhoff-Gustavson normal forms to accurately approximate the spectrum of semiclassical Schrödinger operators, with applications to the near Fermi 1:2 resonance.

Contribution

It extends the Birkhoff-Gustavson normal form to near resonant cases, providing explicit formulas for spectral approximation in semiclassical Schrödinger operators.

Findings

Formulas for spectral approximation in near resonant cases

Explicit expressions for near Fermi 1:2 resonance

Numerical computations validating the approach

Abstract

The concept of near resonances for harmonic approximations of semiclassical Schr\"odinger operators is introduced and explored. Combined with a natural extension of the Birkhoff-Gustavson normal form, we obtain formulas for approaching the discrete spectrum of such operators which are both accurate and easy to implement. We apply the theory to the physically important case of the near Fermi 1:2 resonance, for which we propose explicit expressions and numerical computations.