Trace formula for Sturm-Liouville operators with singular potentials

Suppose that u(chi) is a function of bounded variation on the closed interval [0, pi], continuous at the endpoints of this interval. Then the Sturm-Liouville operator Sy = -y " + q(chi) with Dirichlet boundary conditions and potential q(chi) = u '(chi) is well defined. (The above relation is understood in the sense of distributions.) In the paper, we prove the trace formula (infinity)Sigma (k=1)(lambda (2)(k) - k(2) + b(2k)) = -1/8 Sigma h(j)(2), b(k) = 1/pi integral (pi)(0) cosk chi du(chi), where the lambda (k) are the eigenvalues of S and h(j) are the jumps of the function u(chi). Moreover, in the case of local continuity of q(chi) at the points 0 and pi the series Sigma (infinity)(k=1) (lambda (k) - k(2)) is summed by the mean-value method, and its sum is equal to - (q(0) + q(pi))/4 - 1/8 Sigma h(j)(2).