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Linear representation of energy-dependent Hamiltonians. (English) Zbl 1161.81355

Summary: Quantum mechanics abounds in models with Hamiltonian operators which are energy-dependent. A linearization of the underlying Schrödinger equation with \(H=H(E)\) is proposed here via an introduction of a doublet of separate energy-independent representatives \(K\) and \(L\) of the respective right and left action of \(H(E)\). Both these new operators are non-Hermitian so that our formalism admits a natural extension to non-Hermitian initial \(H(E)\)s. Its applicability may range from pragmatic phenomenology and variational calculations (where all the subspace-projected effective operators depend on energy by construction) up to perturbation theory and quasi-exact constructions.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q40 PDEs in connection with quantum mechanics
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[1] Hilgevoord, J., Am. J. Phys., 70, 301 (2002), with further references
[2] Dittrich, J.; Duclos, P., J. Phys. A, 35, 8213 (2002), with further references · Zbl 1040.35081
[3] Navrátil, P.; Geyer, H. B.; Kuo, T. T.S., Phys. Lett. B, 315, 1 (1993), with further references
[4] Fernández, F. M., Introduction to Perturbation Theory in Quantum Mechanics (2001), CRC Press: CRC Press Boca Raton, FL · Zbl 1048.81021
[5] H. Bı́la, Pseudo-Hermitian Hamiltonians in quantum theory, Diploma Thesis, Charles University, Prague, April 2004, Chapter 2 (in Czech), unpublished; H. Bı́la, Pseudo-Hermitian Hamiltonians in quantum theory, Diploma Thesis, Charles University, Prague, April 2004, Chapter 2 (in Czech), unpublished
[6] Bydžovský, P.; Sotona, M., Czech. J. Phys., 48, 903 (1998), with references
[7] Znojil, M.; Bı́la, H.; Jakubský, V.
[8] Friedman, E.; Gal, A.; Mareš, J., Nucl. Phys. A, 625, 272 (1997), with references
[9] Lichard, P., Phys. Rev. D, 60, 053007 (1999)
[10] Cooper, E. D.; Jennings, B. K.; Mareš, J., Nucl. Phys. A, 580, 419 (1994)
[11] Messiah, A., Quantum Mechanics II (1961), North-Holland: North-Holland Amsterdam
[12] Weigert, S., Czech. J. Phys., 54, 147 (2004)
[13] Ushveridze, A. G., Quasi-Exactly Solvable Models in Quantum Mechanics (1994), IOP: IOP Bristol · Zbl 0834.58042
[14] Singh, V.; Biswas, S. N.; Datta, K., Phys. Rev. D, 18, 1901 (1978)
[15] Cannata, F.; Ioffe, M.; Roychoudhury, R.; Roy, P., Phys. Lett. A, 281, 305 (2001) · Zbl 0984.81041
[16] M. Znojil, New types of solvability in PT symmetric quantum theory, in: Proceedings of Workshop on Superintegrability in Classical and Quantum Systems, 16-21 September, 2002, Montreal, Canada, in press; M. Znojil, New types of solvability in PT symmetric quantum theory, in: Proceedings of Workshop on Superintegrability in Classical and Quantum Systems, 16-21 September, 2002, Montreal, Canada, in press
[17] Azizov, T. Ya.; Iokhvidov, I. S., Linear Operators in Spaces with Indefinite Metric (1989), Wiley: Wiley Chichester · Zbl 0714.47028
[18] Scholtz, F. G.; Geyer, H. B.; Hahne, F. J.W., Ann. Phys. (N.Y.), 213, 74 (1992)
[19] Ramirez, A.; Mielnik, B., Rev. Mexicana Fis., 49, 2, 130 (2003), giving a concise history of the subject
[20] Heiss, W. D.; Harney, H. L., Eur. Phys. J. D, 17, 149 (2001)
[21] Bender, C. M.; Meisinger, P. N.; Wang, Q., J. Phys. A: Math. Gen., 36, 1973 (2003)
[22] Lévai, G.; Cannata, F.; Ventura, A., J. Phys. A: Math. Gen., 35, 5041 (2002)
[23] Mostafazadeh, A., Phys. Lett. A, 320, 375 (2004) · Zbl 0971.82029
[24] Czech. J. Phys., 54, 1-156 (2004), Collection of all 20 papers in Nr. 1 of
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