Exact sum rules for inhomogeneous systems containing a zero mode

• We discuss the sum rules of the eigenvalues of inhomogeneous systems containing a zero mode.
• We derive the explicit expressions for sum rules of order one and two.
• We perform accurate numerical tests of these results for three examples.

We show that the formulas for the sum rules for the eigenvalues of inhomogeneous systems that we have obtained in two recent papers are incomplete when the system contains a zero mode. We prove that there are finite contributions of the zero mode to the sum rules and we explicitly calculate the expressions for the sum rules of order one and two. The previous results for systems that do not contain a zero mode are unaffected.