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Mat. Zametki, 2001, Volume 69, Issue 3, Pages 427–442 (Mi mz515)  

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


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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
\by A.~M.~Savchuk, A.~A.~Shkalikov
\paper Trace Formula for Sturm--Liouville Operators with Singular Potentials
\jour Mat. Zametki
\yr 2001
\vol 69
\issue 3
\pages 427--442
\jour Math. Notes
\yr 2001
\vol 69
\issue 3
\pages 387--400

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    2. Djakov, P, “Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices”, Journal of Approximation Theory, 139:1–2 (2006), 293  crossref  mathscinet  zmath  isi  scopus  scopus
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    21. Hira F., “The Regularized Trace of Sturm-Liouville Problem With Discontinuities At Two Points”, Inverse Probl. Sci. Eng., 25:6 (2017), 785–794  crossref  mathscinet  zmath  isi  scopus  scopus
    22. Kanguzhin B.E., “Weinstein Criteria and Regularized Traces in the Case of Transverse Vibrations of An Elastic String With Springs”, Differ. Equ., 54:1 (2018), 7–12  crossref  mathscinet  zmath  isi  scopus  scopus
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