Trace Formula for Sturm--Liouville Operators with Singular Potentials

A. M. Savchuk ; A. A. Shkalikov

Matematičeskie zametki, Tome 69 (2001) no. 3, pp. 427-442

Résumé

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 $$ -\frac{(q(0)+q(\pi))}4-\frac 18\sum h_j^2. $$