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Isospectral Hamiltonians from Moyal products. (English) Zbl 1118.81032

Summary: Recently Scholtz and Geyer proposed a very efficient method to compute metric operators for non-Hermitian Hamiltonians from Moyal products. We develop these ideas further and suggest to use a more symmetrical definition for the Moyal products, because they lead to simpler differential equations. In addition, we demonstrate how to use this approach to determine the Hermitian counterpart for a pseudo-Hermitian Hamiltonian. We illustrate our suggestions with the explicityly solvable example of the \(-x^4\)-potential and the ubiquitous harmonic oscillator in a complex cube potential.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products
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[1] C.M. Bender and S. Boettcher: Phys. Rev. Lett. 80 (1998) 5243. · Zbl 0947.81018 · doi:10.1103/PhysRevLett.80.5243
[2] M. Znojil: Phys. Lett. A 259 (1999) 220; 264 (1999) 108. · Zbl 0948.81535 · doi:10.1016/S0375-9601(99)00429-6
[3] B. Bagchi, C. Quesne, and M. Znojil: Mod. Phys. Lett. A 16 (2001) 2047. · Zbl 1138.81375 · doi:10.1142/S0217732301005333
[4] S. Weigert: Phys. Rev. A 68 (2003) 062111(4).
[5] C.M. Bender, D.C. Brody, and H.F. Jones: Phys. Rev. Lett. 89, 270401(4) (2002).
[6] C.M. Bender, D.C. Brody, and H.F. Jones: Phys. Rev. D 70 (2004) 025001(19).
[7] A. Mostafazadeh: J. Math. Phys. 43 (2002) 205; 2814; 3944; J. Phys. A 36 (2003) 7081. · Zbl 1059.81070 · doi:10.1063/1.1418246
[8] S. Weigert: J. Phys. B 5 (2003) S416 (2003).
[9] F.G. Scholtz, H.B. Geyer, and F.J.W. Hahne: Ann. Phys. 213 (1992) 74. · Zbl 0749.47041 · doi:10.1016/0003-4916(92)90284-S
[10] A. Mostafazadeh: J. Phys. A 38 (2005) 6557. · Zbl 1072.81020 · doi:10.1088/0305-4470/38/29/010
[11] C. Figueira de Morisson Faria and A. Fring: J. Phys. A 39 (2006) 9269. · Zbl 1095.81026 · doi:10.1088/0305-4470/39/29/018
[12] F.G. Scholtz and H.B. Geyer: Phys. Lett. B 634 (2006) 84; J. Phys. A 39 (2006) 10189. · Zbl 1117.81062
[13] C. Figueira de Morisson Faria and A. Fring: in preparation.
[14] J.E. Moyal: Proc. Cambridge Phil. Soc. 45 (1949) 99. · doi:10.1017/S0305004100000487
[15] D.B. Fairlie: Mod. Phys. Lett. A 13 (1998) 263. · doi:10.1142/S0217732398000322
[16] N. Seiberg and E. Witten: JHEP 09 (1999) 032. · Zbl 0957.81085 · doi:10.1088/1126-6708/1999/09/032
[17] A. Dimakis and F. Müller-Hoissen: Int. J. Mod. Phys. B 14 (2000) 2455. · Zbl 1073.37533 · doi:10.1142/S0217979200001977
[18] M.T. Grisaru and S. Penati: Nucl. Phys. B 655 (2003) 250. · Zbl 1009.81074 · doi:10.1016/S0550-3213(03)00064-6
[19] I. Cabrera-Carnero and M. Moriconi: Nucl. Phys. B 673 (2003) 437. · Zbl 1058.81722 · doi:10.1016/j.nuclphysb.2003.09.014
[20] O. Lechtenfeld, L. Mazzanti, S. Penati, A.D. Popov, and L. Tamassia: Nucl. Phys. B 705 (2005) 477. · Zbl 1119.81331 · doi:10.1016/j.nuclphysb.2004.10.050
[21] D.B. Fairlie: J. of Chaos, Solitons Fractals 10 (1999) 365. · Zbl 0997.81048 · doi:10.1016/S0960-0779(98)00158-1
[22] R. Carroll: North-Holland Mathematics Studies, Vol. 186, Elsevier, Amsterdam, 2000.
[23] H.F. Jones and J. Mateo: Phys. Rev. D 73 (2006) 085002.
[24] H. Jones: J. Phys. A 38 (2005) 1741. · Zbl 1069.81018 · doi:10.1088/0305-4470/38/8/010
[25] A. Mostafazadeh: J. Math. Phys. 47 (2006) 072103.
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