The quantum mechanical atomic kinetic energy and momentum operators commute, and essentially are powers of the gradient operator: it seems that any power of the gradient commutes with any other, and thus would comprise an infinite set of commuting operators that would have a common eigenfunction.
My question more succinctly is to determine what other operators commute with these "gradient" operators, i.e., what are the other commuting operators (besides the identity operator) that with these "gradient" operators form a complete set of commuting operators?
The operators being used are the atomic operators using hydrogenic atomic quantum numbers (to start).
This is involved in research that is seeking a common, closed-form eigenvector solution to a set of differential eigenvalue equations (each equation corresponding to an operator) that explicitly includes electron-electron interactions, including a fundamental wave-mechanical spin-dependent model for electron correlation.
I cannot find that the kinetic and momentum operators alone comprise a complete set, moreover, when I try to solve the corresponding set of differential eigenvalue equations, I get a hodgepodge of functions with coefficients, which I cannot otherwise but arbitrarily evaluate. This problem might be resolved if there was at least another operator that would add another equation to the set that would either reduce the array of possible functions that are solutions, or resolve these coefficients.