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This article is cited in 25 scientific papers (total in 25 papers)
Trace Formula for Sturm–Liouville Operators with Singular Potentials
A. M. Savchuka, A. A. Shkalikovab a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Pohang University of Science and Technology
Abstract:
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
$$
-\frac{(q(0)+q(\pi))}4-\frac 18\sum h_j^2.
$$
DOI:
https://doi.org/10.4213/mzm515
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English version:
Mathematical Notes, 2001, 69:3, 387–400
Bibliographic databases:
UDC:
517.9+517.43 Received: 08.09.2000
Citation:
A. M. Savchuk, A. A. Shkalikov, “Trace Formula for Sturm–Liouville Operators with Singular Potentials”, Mat. Zametki, 69:3 (2001), 427–442; Math. Notes, 69:3 (2001), 387–400
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/mz515https://doi.org/10.4213/mzm515 http://mi.mathnet.ru/eng/mz/v69/i3/p427
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