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# Sturm-Liouville operators with matrix distributional coefficients

### Abstract

The paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ with the matrix-valued coefficients $p,q$ such that $$q=Q', \quad p^{-1},\, p^{-1}Q, \,\, Qp^{-1}, \,\, Qp^{-1}Q \in L_1,$$ where the derivative of the function $Q$ is understood in the sense of distributions. Due to a suitable regularization, the corresponding operators are correctly defined as quasi-differentials. Their resolvent convergence is investigated and all self-adjoint, maximal dissipative, and maximal accumulative extensions are described in terms of homogeneous boundary conditions of the canonical form.

Key words: Sturm-Liouville problem, matrix quasi-differential operator, singular coefficients, resolvent approximation, self-adjoint extension.

### Article Information

 Title Sturm-Liouville operators with matrix distributional coefficients Source Methods Funct. Anal. Topology, Vol. 23 (2017), no. 1, 51-59 MathSciNet MR3632388 Milestones Received 25/10/2016; Revised 23/11/2016 Copyright The Author(s) 2017 (CC BY-SA)

### Authors Information

Alexei Konstantinov
Taras Shevchenko National University of Kyiv, 64 Volodymyrs’ka, Kyiv, 01601, Ukraine

Oleksandr Konstantinov
Livatek Ukraine LLC, 42 Holosiivskyi Ave., Kyiv, 03039, Ukraine

### Citation Example

Alexei Konstantinov and Oleksandr Konstantinov, Sturm-Liouville operators with matrix distributional coefficients, Methods Funct. Anal. Topology 23 (2017), no. 1, 51-59.

### BibTex

@article {MFAT947,
AUTHOR = {Konstantinov, Alexei and Konstantinov, Oleksandr},
TITLE = {Sturm-Liouville operators with matrix distributional coefficients},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {23},
YEAR = {2017},
NUMBER = {1},
PAGES = {51-59},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=947},
}

### References

1. S. Albeverio, Friedrich Gesztesy, Raphael Hoegh-Krohn, and Helge Holden, Solvable models in quantum mechanics, Texts and Monographs in Physics, Springer-Verlag, New York, 1988.  MathSciNet CrossRef
2. S. Albeverio and P. Kurasov, Singular perturbations of differential operators, London Mathematical Society Lecture Note Series, vol. 271, Cambridge University Press, Cambridge, 2000.  MathSciNet CrossRef
3. C. Bennewitz and W. N. Everitt, On second-order left-definite boundary value problems, Ordinary differential equations and operators (Dundee, 1982), Lecture Notes in Math., vol. 1032, Springer, Berlin, 1983, pp. 31-67.  MathSciNet CrossRef
4. V. M. Bruk, A certain class of boundary value problems with a spectral parameter in the boundary condition, Mat. Sb. (N.S.) 100 (142) (1976), no. 2, 210-216.  MathSciNet
5. Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, Alexander Sakhnovich, and Gerald Teschl, Inverse spectral problems for Schrodinger-type operators with distributional matrix-valued potentials, Differential Integral Equations 28 (2015), no. 5-6, 505-522.  MathSciNet
6. Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl, Supersymmetry and Schrodinger-type operators with distributional matrix-valued potentials, J. Spectr. Theory 4 (2014), no. 4, 715-768.  MathSciNet CrossRef
7. W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, Mathematical Surveys and Monographs, vol. 61, American Mathematical Society, Providence, RI, 1999.  MathSciNet
8. Hilbert Frentzen, Equivalence, adjoints and symmetry of quasidifferential expressions with matrix-valued coefficients and polynomials in them, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), no. 1-2, 123-146.  MathSciNet CrossRef
9. Hilbert Frentzen, Quasi-differential operators in $L^ p$ spaces, Bull. London Math. Soc. 31 (1999), no. 3, 279-290.  MathSciNet CrossRef
10. V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Mathematics and its Applications, vol. 48, Springer Netherlands, 1991. CrossRef
11. Andrii Goriunov and Vladimir Mikhailets, Regularization of singular Sturm-Liouville equations, Methods Funct. Anal. Topology 16 (2010), no. 2, 120-130.  MathSciNet
12. Andrii Goriunov, Vladimir Mikhailets, and Konstantin Pankrashkin, Formally self-adjoint quasi-differential operators and boundary-value problems, Electron. J. Differential Equations (2013), no. 101, 1-16.  MathSciNet
13. A. S. Goryunov and V. A. Mikhailets, Regularization of two-term differential equations with singular coefficients by quasiderivatives, Ukrainian Math. J. 63 (2012), no. 9, 1361-1378.  MathSciNet CrossRef
14. A. S. Horyunov, Convergence and approximation of the Sturm-Liouville operators with potentials-distributions, Ukrainian Math. J. 67 (2015), no. 5, 680-689.  MathSciNet CrossRef
15. Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  MathSciNet
16. T. I. Kodlyuk, V. A. Mikhailets, and N. V. Reva, Limit theorems for one-dimensional boundary-value problems, Ukrainian Math. J. 65 (2013), no. 1, 77-90.  MathSciNet CrossRef
17. O. O. Konstantinov, Two-term differential equations with matrix distributional coefficients, Ukrainian Math. J. 67 (2015), no. 5, 711-722.  MathSciNet CrossRef
18. Aleksey S. Kostenko and Mark M. Malamud, 1-D Schrodinger operators with local point interactions on a discrete set, J. Differential Equations 249 (2010), no. 2, 253-304.  MathSciNet CrossRef
19. K. A. Mirzoev, Sturm-Liouville operators, Trans. Moscow Math. Soc. (2014), 281-299.  MathSciNet CrossRef
20. K. A. Mirzoev and T. A. Safonova, On the deficiency index of the vector-valued Sturm-Liouville operator, Math. Notes 99 (2016), no. 2, 290-303.  MathSciNet CrossRef
21. M. Moller and A. Zettl, Semi-boundedness of ordinary differential operators, J. Differential Equations 115 (1995), no. 1, 24-49.  MathSciNet CrossRef
22. A. M. Savchuk and A. A. Shkalikov, Sturm-Liouville operators with singular potentials, Math. Notes 66 (1999), no. 6, 741-753.  MathSciNet CrossRef
23. D. Shin, Quasi-differential operators in Hilbert space, Mat. Sb. 13 (55) (1943), 39-70 (in Russian).  MathSciNet
24. J. D. Tamarkin, A lemma of the theory of linear differential systems, Bull. Amer. Math. Soc. 36 (1930), no. 2, 99-102.  MathSciNet CrossRef
25. Joachim Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987.  MathSciNet CrossRef
26. A. Zettl, Formally self-adjoint quasi-differential operators, Rocky Mountain J. Math. 5 (1975), 453-474.  MathSciNet