Nonstandard Sturm-Liouville difference operator

V.E. Lyantse; Yu.M. Yavorsky

A nonstandard ordinary singular Sturm-Liouville operator $L$ in finite differences is considered. The expansion in principal functions of $L$ is described. The results are similar to the ones for the differential operator of Naimark, though $L$ is an operator in (hyper) finite space.

1. . M.A. Naimark Investigation of the spectrum and expansion in eigenfunctions of a non-selfadjoint differential operator of the 2nd order on the half-axis // Trudy Moskow. Mat. Obshch. V. 3 p.181–270–1954Russian

2. V.E. Lyantse The non-selfadjoint differential operator of the second order on the half-axis. Appendix 2M.A. Naimark, Linear differential operators, part 2. , F. Ungar Publishing CO,addr New Yorkp. 292–331–1968

3. V.A. Marchenko Eigenfunction expansions for non-selfadjoint singular differential operators of the second order// Matem. Sb. V. 52 p. 739–788– 1960Russian

4. N. Dunford and J. Schwartz Linear operators, Part 3. Spectral operators, Wiley-Interscience ,addr New York-London-Sydney-Toronto–1971

5. V.E. Lyantse On the differential operator with the spectral singularities// Mat. Sb. V. 64№4–1964p. 521–561, 65. (1964), no. 1, 47–103 Russian

6. V.E. Lyantse Expansion in eigenfunction // Rev. Roum. Math. Pures et Appl. V. 11№8–1966p. 921–950, 11. (1966), no. 10, 1187–1224

7. V.E. Lyantse Perturbation of continuous spectrum // Dokl. AN SSSR V. 187p. 514–517–1969 Russian

8. V.A. Blashchak On the second–order differential operator on the whole axis with the spectral singularities// Dokl. AN UkrSSR V. 1p. 38–41–1966 Russian

9. E. Nelson Internal Set Theory// Bull. Amer. Math. Soc. V. 83p. 1165–1198–1977

10. E. Nelson Radically Elementary Probability Theory , Princeton Univercity Press–1987

11. F. Diener Cours d'analyse non standard, Offices des Publications Universitaires,addr Alger–1983

12. R. Lutz, M. Goze Non standard analysis: a practical guide with applications (Lect. Notes in Math. V. 881) , Springer Verlag,addr Berlin et New Jork–1981

13. P. Cartier Perturbations singulièrs et analyse non standard// Astérisque, Séminaire Bourbaki V. 92–93–1982

14. G. Reeb Mathématique non standard (essai de vulgarisation)// Bull. AP–MEP V. 328p. 258–273 –1981

15. W. Lyantse Spectrum and resolvent of finite difference operator// Ukr. Math. J. V. 20№ 4 –1968p. 489–503 21. (1969), no. 4, 461–474

	Nonstandard Sturm-Liouville difference operator
Author	V.E. Lyantse, Yu.M. Yavorsky Department of Mechanics and Mathematics,Lviv State University,Universytetska str, 1,290602, Lviv, Ukraine
Abstract	A nonstandard ordinary singular Sturm-Liouville operator $L$ in finite differences is considered. The expansion in principal functions of $L$ is described. The results are similar to the ones for the differential operator of Naimark, though $L$ is an operator in (hyper) finite space.
Keywords	nonstandard Sturm-Liouville operator, finite differences, principal functions expansion
DOI	doi:10.30970/ms.10.1.54-68
Reference	1. . M.A. Naimark Investigation of the spectrum and expansion in eigenfunctions of a non-selfadjoint differential operator of the 2nd order on the half-axis // Trudy Moskow. Mat. Obshch. V. 3 p.181–270–1954Russian 2. V.E. Lyantse The non-selfadjoint differential operator of the second order on the half-axis. Appendix 2M.A. Naimark, Linear differential operators, part 2. , F. Ungar Publishing CO,addr New Yorkp. 292–331–1968 3. V.A. Marchenko Eigenfunction expansions for non-selfadjoint singular differential operators of the second order// Matem. Sb. V. 52 p. 739–788– 1960Russian 4. N. Dunford and J. Schwartz Linear operators, Part 3. Spectral operators, Wiley-Interscience ,addr New York-London-Sydney-Toronto–1971 5. V.E. Lyantse On the differential operator with the spectral singularities// Mat. Sb. V. 64№4–1964p. 521–561, 65. (1964), no. 1, 47–103 Russian 6. V.E. Lyantse Expansion in eigenfunction // Rev. Roum. Math. Pures et Appl. V. 11№8–1966p. 921–950, 11. (1966), no. 10, 1187–1224 7. V.E. Lyantse Perturbation of continuous spectrum // Dokl. AN SSSR V. 187p. 514–517–1969 Russian 8. V.A. Blashchak On the second–order differential operator on the whole axis with the spectral singularities// Dokl. AN UkrSSR V. 1p. 38–41–1966 Russian 9. E. Nelson Internal Set Theory// Bull. Amer. Math. Soc. V. 83p. 1165–1198–1977 10. E. Nelson Radically Elementary Probability Theory , Princeton Univercity Press–1987 11. F. Diener Cours d'analyse non standard, Offices des Publications Universitaires,addr Alger–1983 12. R. Lutz, M. Goze Non standard analysis: a practical guide with applications (Lect. Notes in Math. V. 881) , Springer Verlag,addr Berlin et New Jork–1981 13. P. Cartier Perturbations singulièrs et analyse non standard// Astérisque, Séminaire Bourbaki V. 92–93–1982 14. G. Reeb Mathématique non standard (essai de vulgarisation)// Bull. AP–MEP V. 328p. 258–273 –1981 15. W. Lyantse Spectrum and resolvent of finite difference operator// Ukr. Math. J. V. 20№ 4 –1968p. 489–503 21. (1969), no. 4, 461–474
Pages	54-68
Volume	10
Issue	1
Year	1998
Journal	Matematychni Studii
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