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

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