|1Goriunov, AS |
1Institute of Mathematics of the NAS of Ukraine, Kyiv
|Dopov. Nac. akad. nauk Ukr. 2020, 7:10-16|
The paper investigates spectral properties of multi-interval Sturm–Liouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of boundary conditions are given. Sufficient conditions for the resolvents of these operators to be operators of the trace class and for the systems of root functions to be complete are found. The results are new for one-interval boundary-value problems as well.
|Keywords: Sturm–Liouville operator; multi-interval boundary value problems; distributional coefficients; maximal dissipative extension; completeness of root functions|
1. Goriunov, A. S. & Mikhailets, V. A. (2010). Regularization of singular Sturm—Liouville equations. Meth. Funct. Anal. Topol., 16, No. 2, pp. 120-130.
2. Goriunov, A. S., Mikhailets, V. A. & Pankrashkin, K. (2013). Formally self-adjoint quasi-differential operators and boundary-value problems. Electron. J. Diff. Equ., No. 101, pp. 1-16.
3. Mirzoev, K. A. & Shkalikov, A. A. (2016). Differential operators of even order with distribution coefficients. Math. Notes, 99, No. 5, pp. 779-784.
4. Eckhardt, J., Gesztesy, F., Nichols, R. & Teschl, G. (2013). Weyl—Titchmarsh theory for Sturm—Liouville operators with distributional coefficients. Opusc. Math., 33, No. 3, pp. 467-563.
5. Goriunov, A. S. & Mikhailets, V. A. (2012). Regularization of two-term differential equations with singular coefficients by quasiderivatives. Ukr. Math. J., 63, No. 9, pp. 1361-1378.
6. Everitt, W. N. & Zettl, A. (1986). Sturm—Liouville differential operators in direct sum spaces. Rocky Mountain J. Math., 16, No. 3., pp. 497-516.
7. Everitt, W. N. & Zettl, A. (1992). Quasi-differential operators generated by a countable number of expressions on the real line. Proc. London Math. Soc., 64, No. 3, pp. 524-544.
8. Sokolov, M. S. (2006). Representation results for operators generated by a quasi-differential multi-interval system in a Hilbert direct sum space. Rocky Mt. J. Math., 36, No. 2, pp. 721-739.
9. Goriunov, A. S. (2014). Multi-interval Sturm—Liouville boundary-value problems with distributional potentials. Dopov. Nac. akad. nauk Ukr., No. 7, pp. 43-47.
10. Zettl, A. (1975). Formally self-adjoint quasi-differential operators. Rocky Mt. J. Math., 5, No. 3, pp. 453-474.
11. Everitt, W. N. & Markus, L. (1999). Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators. Providence: American Mathematical Society.
12. Kochubei, A.N. (1975). Extensions of symmetric operators and of symmetric binary relations. Math. Notes., 17, No. 1, pp. 25-28.
13. Gorbachuk, V. I. & Gorbachuk, M. L. (1991). Boundary value problems for operator differential equations. Dordrecht: Kluwer Academic Publishers Group.
14. Bruk, V. M. (1976). A certain class of boundary value problems with a spectral parameter in the boundary condition. Mat. Sb. (N.S.), 100, No. 2, pp. 210-216 (in Russian).
15. Goriunov, A. S. (2015). Convergence and approximation of the Sturm—Liouville operators with potentialsdistributions. Ukr. Math. J., 67, No. 5, pp. 680-689.