Some Remarks About Semiclassical Trace Invariants and Quantum Normal Forms

TL;DR

This paper investigates the relationship between semi-classical and quantum Birkhoff canonical forms for Schrödinger operators, offering a new operator-theoretic derivation and explicit connection between the two forms.

Contribution

It provides a novel non-symbolic operator-theoretic derivation of the quantum Birkhoff canonical form and an explicit method to relate it to the semi-classical BCF.

Findings

Derived a non-symbolic operator-theoretic approach for quantum BCF

Established an explicit recipe connecting semi-classical and quantum BCF

Enhanced understanding of the link between semi-classical and quantum spectral invariants

Abstract

In this paper we explore the connection between semi-classical and quantum Birkhoff canonical forms (BCF) for Schrodinger operators. In particular we give a "non-symbolic" operator theoretic derivation of the quantum Birkhoff canonical form and provide an explicit recipe for expressing the quantum BCF in terms of the semi-classical BCF.