Trace Formula for Sturm–Liouville Operators with Singular Potentials

Suppose that $u(x)$ is a function of bounded variation on the closed interval $[0,\pi]$, continuous at the endpoints of this interval. Then the Sturm–Liouville operator $Sy=-y"+q(x)$ with Dirichlet boundary conditions and potential $q(x)=u'(x)$ is well defined. (The above relation is understood in the sense of distributions.) In the paper, we prove the trace formula
$$ \sum_{k=1}^\infty(\lambda_k^2-k^2+b_{2k}) =-\frac 18\sum h_j^2, $$
where the $\lambda_k$ are the eigenvalues of $S$ and $h_j$ are the jumps of the function $u(x)$. Moreover, in the case of local continuity of $q(x)$ at the points 0 and $\pi$ the series $\sum_{k=1}^\infty(\lambda_k-k^2)$ is summed by the mean-value method, and its sum is equal to
$$ -\frac{(q(0)+q(\pi))}4-\frac 18\sum h_j^2. $$