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

This article is cited in 24 scientific papers (total in 24 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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    This publication is cited in the following articles:
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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
    3. Yu. V. Pokornyi, M. B. Zvereva, S. A. Shabrov, “Sturm–Liouville oscillation theory for impulsive problems”, Russian Math. Surveys, 63:1 (2008), 109–153  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. N. M. Aslanova, “A trace formula of a boundary value problem for the operator Sturm–Liouville equation”, Siberian Math. J., 49:6 (2008), 959–967  mathnet  crossref  mathscinet  isi  elib
    5. Djakov, P, “Spectral gap asymptotics of one-dimensional Schrodinger operators with singular periodic potentials”, Integral Transforms and Special Functions, 20:3–4 (2009), 265  crossref  mathscinet  zmath  isi  scopus  scopus
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    8. Yang Ch.F., “New trace formulae for a quadratic pencil of the Schroumldinger operator”, Journal of Mathematical Physics, 51:3 (2010), 033506  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
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    11. Djakov P. Mityagin B., “Fourier Method for One-Dimensional Schrodinger Operators with Singular Periodic Potentials”, Topics in Operator Theory, Vol 2: Systems and Mathematical Physics, Operator Theory Advances and Applications, 203, ed. Ball J. Bolotnikov V. Helton J. Rodman L. Spitkovsky I., Birkhauser Verlag Ag, 2010, 195–236  crossref  mathscinet  isi
    12. È. F. Akhmerova, “Asymptotics of the Spectrum of Nonsmooth Perturbations of Differential Operators of Order $2m$”, Math. Notes, 90:6 (2011), 813–823  mathnet  crossref  crossref  mathscinet  isi
    13. Yang Ch.-F., “Regularized Trace for Sturm-Liouville Differential Operator on a Star-Shaped Graph”, Complex Anal. Oper. Theory, 7:4 (2013), 1185–1196  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    14. Djakov P. Mityagin B., “Equiconvergence of Spectral Decompositions of Hill-Schrodinger Operators”, J. Differ. Equ., 255:10 (2013), 3233–3283  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. M. E. Akhymbek, D. B. Nurakhmetov, “Pervyi regulyarizovannyi sled operatora dvukratnogo differentsirovaniya na prokolotom otrezke”, Sib. elektron. matem. izv., 11 (2014), 626–633  mathnet
    16. Eckhardt J. Gesztesy F. Nichols R. Teschl G., “Supersymmetry and Schrodinger-Type Operators With Distributional Matrix-Valued Potentials”, J. Spectr. Theory, 4:4 (2014), 715–768  crossref  mathscinet  zmath  isi  scopus  scopus
    17. Nazarov A.I. Stolyarov D.M. Zatitskiy P.B., “The Tamarkin Equiconvergence Theorem and a First-Order Trace Formula For Regular Differential Operators Revisited”, J. Spectr. Theory, 4:2 (2014), 365–389  crossref  mathscinet  zmath  isi  scopus  scopus
    18. Yang Ch.-F., “Traces of Sturm-Liouville Operators With Discontinuities”, Inverse Probl. Sci. Eng., 22:5 (2014), 803–813  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Luger A., Teschl G., Woehrer T., “Asymptotics of the Weyl function for Schr?dinger operators with measure-valued potentials”, Mon.heft. Math., 179:4 (2016), 603–613  crossref  mathscinet  zmath  isi  scopus
    20. Wang Yu.-p. Koyunbakan H. Yang Ch.-f., “A trace formula for integro-differential operators on the finite interval”, Acta Math. Appl. Sin.-Engl. Ser., 33:1 (2017), 141–146  crossref  mathscinet  zmath  isi  scopus
    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
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