The symbol of a function of an operator

TL;DR

This paper derives an explicit formula for the symbol of a function of an operator and applies it to compute Bohr-Sommerfeld quantization rules at order 4 in hbar, advancing operator symbol calculus.

Contribution

It provides a new explicit formula for the symbol of a function of an operator, enabling precise quantization calculations.

Findings

Explicit formula for the symbol of f() in terms of A.

Calculation of Bohr-Sommerfeld quantization rules at order 4 in bar.

Enhanced understanding of operator symbol calculus.

Abstract

We give an explicit formula for the symbol of a function of an operator. Given an operator {\hat A} on L^2(R^N) with symbol A, and a function f, we obtain the symbol of f(\hat A) in terms of A. As an application, Bohr-Sommerfeld quantization rules are explicitely calculated at order 4 in hbar.