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Trace formula for Sturm-Liouville operators with singular potentials. (English. Russian original) Zbl 1005.34077
Math. Notes 69, No. 3, 387-400 (2001); translation from Mat. Zametki 69, No. 3, 427-442 (2001).
The trace formula for the Sturm-Liouville operator $$S$$ on $$L^2(0,\pi)$$ generated by $$-y''+q(x)y$$ with the Dirichlet boundary conditions $$y(0)=y(\pi)=0$$ is considered. For sufficiently regular potential, this formula is well-known. Here, the case $$q=u'$$ (in the sense of distributions) is investigated, where $$u$$ is of bounded variation. If $$(\lambda _k)$$ denotes the sequence of eigenvalues and $$h_j$$ the jumps of $$u$$, the trace formula $\sum_{k=1}^\infty (\lambda _k-k^2+b_{2k}) =-\frac 18\sum h_j^2,\quad b_k=\frac 1\pi\int_0^\pi \cos kx du(x),$ is shown. If, additionally, $$q$$ is continuous at $$0$$ and $$\pi$$, then the mean-value of the series $$\sum_{k=1}^\infty (\lambda _k-k^2)$$ is $$-\frac{q(0)+q(\pi)}4-\frac 18\sum h_j^2$$.

##### MSC:
 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators 34B24 Sturm-Liouville theory
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