Trace formula for the Sturm-Liouville operator with singularity

Abstract

Let mu(1), mu(2),..., mu(n),... be the Dirichlet spectrum of the operator -d(2)/dx(2) + q(x) acting on L-2(0,pi). In the special case where q(x) = 0, mu(n) = n(2). In the [1] and others discovered the asymptotic formula mu(n), = n(2) + 1/pi integral(0)(pi) q(x)dx +O(n(-2)) and the trace formula Sigma(n) [mu(n) - n(2)] = q(0) + q(pi)/4 , provided that integral(0)(pi) q(x)dx = 0, where q(x) is an element of C-2[0,pi]. There are beautiful formulas with applications for example in solving inverse problems. In this work, the above mentioned problem has been studied for a Sturm-Liouville operator with A/x (A is real) singularity at x = 0.