On new classes of Hermitian exactly solvable matrix Schrödinger
operators (in Ukrainian)

A. O. Abramenko; S. V. Spichak

Five multi-parameter families of Hermitian exactly solvable matrix Schrödinger operators in one variable was constructed. For this we use a representation of the four-dimensional Lie algebra as well as invariant space under its operators and additional operators which act in this space.

1. Zaslavskii O. B., Ulyanov V. V. New classes of exact solutions of the Schrödinger equation and potential-field description of spin systems // Soviet Phys. JETP. – 1984. – V.60, № 5. – P.991–996.

2. Turbiner A. V. Quasi-exactly-solvable problems and ${\rm sl}(2)$ algebra // Comm. Math. Phys.~--~1988.~-- V.118, № 3.~--~P.467--474.

3. Shifman M. A. New findings in quantum mechanics (partial algebraization of the spectral problem) // Internat. J. Modern Phys. A. – 1989. – V.4, № 12. – P.2897–2952.

4. Shifman M. A., Turbiner A. V. Quantal problems with partial algebraization of the spectrum // Comm. Math. Phys. – 1989. – V.126, № 2. – P.347–365.

5. González-López A., Kamran N., Olver N. Quasi-exactly solvable Lie algebras of differential operators in two complex variables // J. Phys. A. – 1991. – V.24, № 17. – P.3995–4008.

6. González-López A., Kamran N., Olver N. New quasi-exactly solvable Hamiltonians in two dimensions // Comm. Math. Phys. – 1994. – V.159, № 3. – P.503–537.

7. Ushveridze A. G. Quasi–exactly Solvable Models in Quantum Mechanics. – Bristol: IOP Publishing, 1993.

8. Finkel F., González-López A., Rodríguez M. A. Quasi-exactly solvable spin $1/2$ Schr\" odinger operators // Preprint hep-th/9509057, 1995.

9. Zhdanov R. Z. On algebraic classification of quasi-exactly solvable matrix models // J. Phys. A. – 1997. – V.30, № 24. – P.8761–8770.

10. Turbiner A. V. Lie algebras and linear operators with invariant subspaces. Lie algebras, cohomology and new applications to quantum mechanics // Contemp. Math., Amer. Math. Soc. – 1994. – V.160. – P.263–310.

11. Brihaye Y., Giller S., Gonera C., Kosiński P. The structure of quasi-exactly solvable systems // J. Math. Phys. – 1995. – V.36, № 8. – P.4340–4349.

12. Finkel F., González-López A., Rodríguez M. A. Quasi-exactly solvable spin $1/2$ Schr\" odinger operators http://194.67.23.76// J. Phys. A.~--~1997.~--~V.30, № 19.~--~P.6879--6892.

13. Абраменко А. О. Matrix realizations of four-dimensional Lie algebras and corresponding invariant spases // Вісник Київського університету, Сер.: фіз.-мат. науки. – 1999. – № 3. – P.11–17.

14. Spichak S., Zhdanov R. On algebraic classification of Hermitian quasi-exactly solvable matrix Schrödinger operators on line // J. Phys. A. – 1999. – V.32, № 20. – P.3815–3831.

	On new classes of Hermitian exactly solvable matrix Schrödinger operators (in Ukrainian)
Author	A. O. Abramenko, S. V. Spichak Poltava V. G. Korolenko National Pedagogical University, Institute of Mathematics of the National Academy of Sciences of Ukraine.
Abstract	Five multi-parameter families of Hermitian exactly solvable matrix Schrödinger operators in one variable was constructed. For this we use a representation of the four-dimensional Lie algebra as well as invariant space under its operators and additional operators which act in this space.
Keywords	five multi-parameter families, Hermitian exactly solvable matrix Schrödinger operators, one variable, representation of the four-dimensional Lie algebra, invariant space, additional operators
DOI	doi:10.30970/ms.15.1.44-56
Reference	1. Zaslavskii O. B., Ulyanov V. V. New classes of exact solutions of the Schrödinger equation and potential-field description of spin systems // Soviet Phys. JETP. – 1984. – V.60, № 5. – P.991–996. 2. Turbiner A. V. Quasi-exactly-solvable problems and ${\rm sl}(2)$ algebra // Comm. Math. Phys.~--~1988.~-- V.118, № 3.~--~P.467--474. 3. Shifman M. A. New findings in quantum mechanics (partial algebraization of the spectral problem) // Internat. J. Modern Phys. A. – 1989. – V.4, № 12. – P.2897–2952. 4. Shifman M. A., Turbiner A. V. Quantal problems with partial algebraization of the spectrum // Comm. Math. Phys. – 1989. – V.126, № 2. – P.347–365. 5. González-López A., Kamran N., Olver N. Quasi-exactly solvable Lie algebras of differential operators in two complex variables // J. Phys. A. – 1991. – V.24, № 17. – P.3995–4008. 6. González-López A., Kamran N., Olver N. New quasi-exactly solvable Hamiltonians in two dimensions // Comm. Math. Phys. – 1994. – V.159, № 3. – P.503–537. 7. Ushveridze A. G. Quasi–exactly Solvable Models in Quantum Mechanics. – Bristol: IOP Publishing, 1993. 8. Finkel F., González-López A., Rodríguez M. A. Quasi-exactly solvable spin $1/2$ Schr\" odinger operators // Preprint hep-th/9509057, 1995. 9. Zhdanov R. Z. On algebraic classification of quasi-exactly solvable matrix models // J. Phys. A. – 1997. – V.30, № 24. – P.8761–8770. 10. Turbiner A. V. Lie algebras and linear operators with invariant subspaces. Lie algebras, cohomology and new applications to quantum mechanics // Contemp. Math., Amer. Math. Soc. – 1994. – V.160. – P.263–310. 11. Brihaye Y., Giller S., Gonera C., Kosiński P. The structure of quasi-exactly solvable systems // J. Math. Phys. – 1995. – V.36, № 8. – P.4340–4349. 12. Finkel F., González-López A., Rodríguez M. A. Quasi-exactly solvable spin $1/2$ Schr\" odinger operators http://194.67.23.76// J. Phys. A.~--~1997.~--~V.30, № 19.~--~P.6879--6892. 13. Абраменко А. О. Matrix realizations of four-dimensional Lie algebras and corresponding invariant spases // Вісник Київського університету, Сер.: фіз.-мат. науки. – 1999. – № 3. – P.11–17. 14. Spichak S., Zhdanov R. On algebraic classification of Hermitian quasi-exactly solvable matrix Schrödinger operators on line // J. Phys. A. – 1999. – V.32, № 20. – P.3815–3831.
Pages	44-56
Volume	15
Issue	1
Year	2001
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On new classes of Hermitian exactly solvable matrix Schrödinger operators (in Ukrainian)