Alfonso GraciaSaz
BohrSommerfeld rules and the symbol of a function of an operator
Weyl Quantization provides a correspondence between quantum observables (i.e. operators on a Hilbert space) and classical observables (i.e. ``symbols'' or smooth functions on a Poisson manifold) in the description of a physical system. BohrSommerfeld rules are an asymptotic method to calculate the eigenvalues of an operator via its symbol. To develop BohrSommerfeld rules for an operator, we first need to solve the following problem: given an operator $A$ and a function $f$, what is the symbol of $f(A)$ in terms of the symbol of $A$? We will explain the context of this problem and give a solution in the form of a combinatorial formula ``a la Feynmann'', which has other applications in quantum mechanics.
