I could give a better answer if I actually knew what a resolvent was, but from cursory googling it has something to do with the spectrum of an operator. I’m gonna pretend this question is about eigenvalues of a linear operator on a finite dimensional vector space, and hope that’s close enough that you can translate back to the actual situation.
Eigenvalues don’t behave well with respect to addition of operators, so I wouldn’t expect knowing about each separately would tell you anything about the sum. On the other hand, if your operators commute, you can simultaneously diagonalize them, and then their eigenvalues would pair up and add. Unfortunately I don’t think your operators commute unless dq/dx = 0, so that’s not super helpful.