Oscar Rosas–Ortiz
Departamento de Física, Cinvestav, A.P. 14–740, México 07000, D.F. México
Recibido el 18 de julio de 2005
Aceptado el 14 de marzo de 2005
Abstract
]]> Gamow solutions are used to transform self–adjoint energy operators by means of factorization (supersymmetric) techniques. The transformed non–hermitian operators admit a discrete real spectrum which is occasionally extended by a single complex eigenvalue associated to normalized eigensolutions. These new Hamiltonians are not pseudo–hermitian operators and also differ from those obtained by means of complex–scaling transformations. As an example, Coulomb–like potentials are studied.Keywords: Factorization method, Gamow vectors, Non–hermitian Hamiltonians.
Resumen
El método de factorización es extendido al caso complejo para construir Hamiltonianos no Hermitianos con espectro real. Algunos de los nuevos Hamiltonianos admiten además un eigenvalor complejo con eigenfunción normalizada. Las funciones de transformación usadas son funciones de Gamow. Los nuevos Hamiltonianos no son pseudo–hermitianos y son diferentes también de aquellos obtenidos con el método de dilatación compleja. Se presenta el caso de potenciales Coulombianos como ejemplo.
Descriptores: Método de factorización, vectores de Gamow, Hamiltonianos no Hermitianos.
PACS: 03.65.Ca; 03.65.Ge; 03.65.Fd
DESCARGAR ARTÍCULO EN FORMATO PDF
]]>Acknowledgements
The author is grateful to the organizers of the V Taller de la DGFM–SMF, Morelia Michoacán, México, for the kind invitation. Special thanks are due to U. Nucamendi for the warm hospitality. The author is indebted to M. Lomeli for the manuscript's typing. The support of CONACyT is acknowledged.
References
1. G. Gamow, Z. Phys. 51 (1928) 204. [ Links ]
2. V.A. Fock, Fundamentals of Quantum Mechanics (URSS Publishers, Moscow 1931) translated from Russian by E Yankovsky. [ Links ]
3. G. Breit and E.P. Wigner, Phys. Rev. 49 (1936) 519. [ Links ]
4. W. Heitler and N. Hu, Nature 159 (1947) 776. [ Links ]
5. R. de la Madrid, Quantum Mechanics in Rigged Hilbert Space Language (PhD dissertation, Theoretical Physics Department, Universidad de Valladolid, Spain 2001) [ Links ]
6. A. Bohm, M. Gadella, and G.B. Mainland, Am. J. Phys. 57 (1989) 1103. [ Links ]
7. P.A.M. Dirac, The principles of quantum mechanics (Oxford University Press, London 1958) [ Links ]
8. K. Maurin Generalized eigenfunction expansions and unitary representations of topological groups (Pan Stuvowe Wydawn, Warsav 1968) [ Links ]
9. I.M. Gelfand and N.Y. Vilenkin, Generalized functions V4 (Academic Press, New York 1968) [ Links ]
10. I Aguilar and J.M. Combes, Commun Math. Phys. 22 (1971) 269; [ Links ] E. Balslev and J.M. Combes Commun Math. Phys. 22 (1971) 280. [ Links ]
11. B. Simon, Commun Math. Phys. 27 (1972) 1. [ Links ]
12. B.G. Giraud and K. Kato, Annals. of Phys. 308 (2003) 115–142. [ Links ]
13. E.C.G. Sudarshan, C.B. Chiu, and V. Gorini, Phys. Rev. D 18 (1978) 2914. [ Links ]
14. T. Berggren, Phys. Lett. B 44 (1973) 23. [ Links ]
15. N. Moiseyev, Phys. Rep. 302 (1998) 211. [ Links ]
16. A. Mostafazadeh, J. Math. Phys. 45 (2004) 932. [ Links ]
17. O. Rosas–Ortiz and R. Muñoz, J. Phys. A: Math Gen 36 (2003) 8497. [ Links ]
18. A. Messiah, Quantum Mechanics, Two Volumes Bound as One (Dover Publications, New York 1999) [ Links ]
19. T.E. Hull and L. Infeld, Rev. Mod. Phys. 23 (1951) 21; [ Links ] A.A. Andrianov, N.V. Borisov, and M.V. Ioffe Phy Lett A 105 (1984) 19. [ Links ]
20. J. Negro, L.M. Nieto, and O. Rosas–Ortiz, J. Phys. A: Math. Gen. 33 (2000) 7207. [ Links ]
21. O. Rosas–Ortiz, J. Negro, and L.M. Nieto Rev. Mex. Fís. 49S1 (2003) 88. [ Links ]
22. A.A. Andrianov, N.V. Borisov, M.V. Ioffe, and M.I. Eides, Theor. Math. Phys. 61 (1984) 965; [ Links ] C.V Sukumar, J. Phys. A: Math. Gen. 18 (1985) 2917. [ Links ]
23. R. Muñoz, Complex Supersymmetric Transformations and Exactly Solvable Potentials (PhD dissertation, Physics Department, Cinvestav, Mexico 2004) [ Links ]
24. I.C. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint operators (American Mathematical Society, USA 1969); [ Links ] T. Y Azizov and I.S. Iokhvidov, Linear operators in spaces with an indefinite metric (John Wiley, New York 1989). [ Links ]
25. A. Ramírez and B. Mielnik Rev. Mex. Fís. 49S2 (2003) 130. [ Links ]
26. R. de la Madrid and M. Gadella Am. J. Phys. 70 (2002) 626. [ Links ]
27. B. Mielnik and O. Rosas–Ortiz, J. Phys. A: Math. and Gen. 37 (2004) 10007. [ Links ]
28. B. Mielnik, J. Math. Phys. 25 (1984) 3387; [ Links ] D. J. Fernández Lett. Math. Phys. 8 (1984) 337. [ Links ]
29. D. J. Fernández, M.L. Glasser, and L.M. Nieto Phys. Lett. A 240 (1998) 15; [ Links ] J.O. Rosas–Ortiz, J. Phys. A: Math. Gen. 31 (1998) 10163. [ Links ]
30. D.J. Fernández and H. Rosu, Rev. Mex. Fís. 46 S2 (2000) 153; [ Links ] D.J. Fernández and H. Rosu Phys. Scr. 64 (2001) 177. [ Links ]
31. J. Negro, L.M. Nieto and O. Rosas–Ortiz, J. Math. Phys. 41 (2000) 7964. [ Links ]
32. C.M. Bender, D.C. Brody, and H.F. Jones, Am. J. Phys. 71 (2003) 1095. [ Links ]
33. D.J. Fernández, R. Muñoz, and A. Ramos, Phys. Lett. A 308 (2003) 11. [ Links ] ]]>