Exact sum rules for inhomogeneous strings

•We derive an explicit expression for the sum rules of an inhomogeneous string.•We obtain a diagrammatic representation for the sum rules of a given order.•We obtain precise bounds on the lowest eigenvalue of the string.

We derive explicit expressions for the sum rules of the eigenvalues of inhomogeneous strings with arbitrary density and with different boundary conditions. We show that the sum rule of order NN may be obtained in terms of a diagrammatic expansion, with (N−1)!/2(N−1)!/2 independent diagrams. These sum rules are used to derive upper and lower bounds to the energy of the fundamental mode of an inhomogeneous string; we also show that it is possible to improve these approximations taking into account the asymptotic behavior of the spectrum and applying the Shanks transformation to the sequence of approximations obtained to the different orders. We discuss three applications of these results.