Itzykson-Zuber Integral & Degenerate Eigenvalues

[Alamino]

It seems that according to the formula, when the eigenvalues of one of the matrices inside the trace of the IZ integral are degenerate, the integral diverges.

Is it correct or the formula is different for this case? For instance, suppose the group is U(N) and I want to calculate

$$\int dU \, \exp\left[\mbox{Tr} (U A U^\dagger \sigma^1_z)\right]$$

where A is Hermitean and $$\sigma^1_z$$ is the spin in the Z direction of the first spin in a 2-qubit system, i.e., the direct product of two Z Pauli matrices. Is the integral divergent in this case?

 

[Evalesco]

The integral would diverge in this case if the eigenvalues of U A U^\dagger are degenerate. The formula for calculating the IZ integral does not change in the case of degenerate eigenvalues - it just means that the integral will not converge. If you are looking to calculate this integral, then you need to be careful that you don't end up with any degeneracies in your eigenvalues.