Sums of zeros of eigenfunctions to differential operators with polynomial coefficients

We consider the boundary value problem –ű +p(x)u = λu (0 < x < 1), where p(z) (z ∈ ℂ) is a complex polynomial. The boundary conditions are not assumed to be selfadjoint. Let zk(ϕ), k = 1, 2,… be the zeros of an eigenfunction to the considered problem. Inequalities for the sums [Formula Presented] (j = 1, 2,…) are derived. These inequalities are new even in the selfadjoint case. Applications of the obtained bounds are also discussed. An illustrative example is presented. It shows that the suggested results are sharp.