On spectral properties of the fourth order differential operator with singular coefficients

S.S.Man'ko

A formal fourth order differential operator with a singular coefficient that is a linear combination of the Dirac delta-function and its derivatives is considered. The asymptotic behavior of spectra and eigenfunctions of a family of differential operators with smooth coefficients approximating the singular coefficient is studied. We explore how behavior of eigenvalues and eigenfunctions is influenced by singular coefficients. The limit operator is constructed and is shown to depend on a type of approximation of singular coefficients.

1. S. Albeverio, F. Gesztesy, R. Hegh-Krohn, H. Holden, Solvable models in quantum mechanics. With an appendix by Pavel Exner, RI: AMS Chelsea Publishing, 2005.

2. S. Albeverio, P. Kurasov, Singular perturbations of differential operators and solvable Schrödingerr type operators, Cambridge, Univ. Press, 2000.

3. T. Azizov, I. Iokhvidov, Linear operators in space with an indefinite metric, Pure and Applied Mathematics, Chichester, 1989.

4. M.~Birman, M.~Solomyak, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalization, Adv. Sov. Math. 7 (1991), 1-55.

5. F. Berezin, M. Shubin, The Schrödinger equation, Kluwer Academic Publishers, 1991.

6. F. Berezin, L. Faddeev, A remark on Schrödinger's equation with a singular potential, Sov. Math. Dokl. 2 (1961), 372-375.

7. B. Ćurgus, H. Langer, A Krein space approach to symmetric ordinary differential operators with an indefinite weigth function, J. Diff. Eq. 79 (1989), 31-61.

8. Yu.~Demkov, V.~Ostrovskii, Zero-range potentials and their applications in atomic physics, Leningrad, Leningrad Univ. Press, 1975.

9. N. Goloscshapova, L. Oridoroga, 4-th order differential operator with local point interactions, Ukr. Math. Bull. 4 (2007), 355-369.

10. Yu. Golovaty, S. Man'ko, Schrödinger operator with $\delta'$-potential, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. 5 (2009), 16-21.

11. Yu. Golovaty, S. Man'ko, Solvable models for the Schrödinger operators with $\delta'$-like potentials, Ukr. Mat. Bull. 6 (2009), 179-212.

12. S. Man'ko, Fourth order differential operators with distributions in coefficients, Visn. L'viv. Univ., Ser. Mekh.-Mat. 72 (2010) (accepted).

13. M. Najmark, Linear differential operators, Moscow, Nauka, 1969. (Russian)

14. M. Vishik, L. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Usp. Mat. Nauk 12 (1957), 3-122.

15. A. Yablonski, Differential equations with generalized coefficients, Nonlinear Anal., Theory Methods Appl. 63 (2005), 171-197.

16. S.~Zavalishchin, A.~Sesekin, Dynamic impulse systems: theory and applications, Dordrecht, Kluwer Academic Publishers Group, 1997.

	On spectral properties of the fourth order differential operator with singular coefficients
Author	S.S.Man'ko stepan.manko@gmail.com Department of Mechanics and Mathematics, Ivan Franko National Univetsity of L'viv, 1 Universytets'ka str., 79000 L'viv, Ukraine
Abstract	A formal fourth order differential operator with a singular coefficient that is a linear combination of the Dirac delta-function and its derivatives is considered. The asymptotic behavior of spectra and eigenfunctions of a family of differential operators with smooth coefficients approximating the singular coefficient is studied. We explore how behavior of eigenvalues and eigenfunctions is influenced by singular coefficients. The limit operator is constructed and is shown to depend on a type of approximation of singular coefficients.
Keywords	spectral property, fourth order differential operator, singular coefficient
DOI	doi:10.30970/ms.33.2.173-191
Reference	1. S. Albeverio, F. Gesztesy, R. Hegh-Krohn, H. Holden, Solvable models in quantum mechanics. With an appendix by Pavel Exner, RI: AMS Chelsea Publishing, 2005. 2. S. Albeverio, P. Kurasov, Singular perturbations of differential operators and solvable Schrödingerr type operators, Cambridge, Univ. Press, 2000. 3. T. Azizov, I. Iokhvidov, Linear operators in space with an indefinite metric, Pure and Applied Mathematics, Chichester, 1989. 4. M.~Birman, M.~Solomyak, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalization, Adv. Sov. Math. 7 (1991), 1-55. 5. F. Berezin, M. Shubin, The Schrödinger equation, Kluwer Academic Publishers, 1991. 6. F. Berezin, L. Faddeev, A remark on Schrödinger's equation with a singular potential, Sov. Math. Dokl. 2 (1961), 372-375. 7. B. Ćurgus, H. Langer, A Krein space approach to symmetric ordinary differential operators with an indefinite weigth function, J. Diff. Eq. 79 (1989), 31-61. 8. Yu.~Demkov, V.~Ostrovskii, Zero-range potentials and their applications in atomic physics, Leningrad, Leningrad Univ. Press, 1975. 9. N. Goloscshapova, L. Oridoroga, 4-th order differential operator with local point interactions, Ukr. Math. Bull. 4 (2007), 355-369. 10. Yu. Golovaty, S. Man'ko, Schrödinger operator with $\delta'$-potential, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. 5 (2009), 16-21. 11. Yu. Golovaty, S. Man'ko, Solvable models for the Schrödinger operators with $\delta'$-like potentials, Ukr. Mat. Bull. 6 (2009), 179-212. 12. S. Man'ko, Fourth order differential operators with distributions in coefficients, Visn. L'viv. Univ., Ser. Mekh.-Mat. 72 (2010) (accepted). 13. M. Najmark, Linear differential operators, Moscow, Nauka, 1969. (Russian) 14. M. Vishik, L. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Usp. Mat. Nauk 12 (1957), 3-122. 15. A. Yablonski, Differential equations with generalized coefficients, Nonlinear Anal., Theory Methods Appl. 63 (2005), 171-197. 16. S.~Zavalishchin, A.~Sesekin, Dynamic impulse systems: theory and applications, Dordrecht, Kluwer Academic Publishers Group, 1997.
Pages	173-191
Volume	33
Issue	2
Year	2010
Journal	Matematychni Studii
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