Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/regularized-trace-of-a-sturm-liouville-operator-on-a-curve-with-a-0VdHz4k6yj
References (14)
-
(1951)
Translated under the title: Teoriya dzeta-funktsii Rimana -
(1953)
Translated under the title: Vysshie transtsendentnye funktsii. T. 2. Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonal'nye mnogochleny -
(1995)
Asymptotic behavior of the spectrum and the regularized trace of higher-order singular differential operators -
(1988)
Operatory Shturma–Liuvillya i Diraka (Sturm–Liouville and Dirac Operators) -
Kh. Ishkin (2016)
Localization criterion for the spectrum of the Sturm–Liouville operator on a curveSt Petersburg Mathematical Journal, 28
-
Kh. Ishkin (2013)
On a trivial monodromy criterion for the Sturm-Liouville equationMathematical Notes, 94
-
Kh. Ishkin (2008)
On the uniqueness criterion for solutions of the Sturm-Liouville equationMathematical Notes, 84
-
(1978)
Spetsial’nye funktsii matematicheskoi fiziki (Special Functions of Mathematical Physics) -
Kh. Ishkin (2005)
Necessary Conditions for the Localization of the Spectrum of the Sturm-Liouville Problem on a CurveMathematical Notes, 78
-
E. Titchmarsh, D. Heath-Brown (1987)
The Theory of the Riemann Zeta-Function -
(1953)
One simple identity for the eigenvalues of a second-order differential operator, Dokl -
Kh. Ishkin (2008)
On localization of the spectrum of the problem with complex weightJournal of Mathematical Sciences, 150
-
(1978)
Teoriya analiticheskikh funktsii -
(1953)
On one Gel’fand–Levitan formula, Usp
- Publisher
- Springer Journals
- Copyright
- Copyright © Pleiades Publishing, Ltd. 2020
- ISSN
- 0012-2661
- eISSN
- 1608-3083
- DOI
- 10.1134/S00122661200100018
- Publisher site
- See Article on Publisher Site
Abstract
For a Sturm–Liouville operator on a piecewise smooth curve, we study the effect that thespectrum of a nonintegrable singularity of the potential on the segment joining the endpoints ofthis curve has on the operator asymptotics. It is shown that in the case where the singular pointdoes not give rise to the branching of solutions in its neighborhood (the case of trivialmonodromy), the spectral asymptotics and the formula for the regularized trace look the same asfor the classical Sturm–Liouville operator on a segment with smooth potential. Further, it isshown that in the case of nontrivial monodromy the spectral asymptotics substantially depends onthe commensurability of the parts into which the segment is partitioned by the singular point\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta$$\end{document}: if \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta$$\end{document} is rational, then the spectrum is divided intofinitely many series, each going to infinity along “its own” parabola. In this case, the regularizedtrace formula is significantly more complicated and does not show any similarity with the classicalformula.
Journal
Differential Equations – Springer Journals
Published: Oct 13, 2020