Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries

Journal of Mathematical Physics

Volumn 43, Issue 8, 2002, Pages 3944-3951

  Mostafazadeh, Ali ^a  

^a KOÇ UNIVERSITY   (Turkey)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035981741     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1489072     Document Type: Article

References (39)

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2. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
3. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
4. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
5. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
6. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
7. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
8. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
9. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
10. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
11. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
12. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
13. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
14. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
15. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
16. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
17. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
18. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
19. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
20. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
21. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998); F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil, J. Phys. A 31, 10105 (1998); F. Cannata, G. Junker, and J. Trost, Phys. Lett. A 246, 219 (1998); C.M. Bender, S. Boettcher, and P.N. Meisenger, J. Math. Phys. 40, 2201 (1999); C.M. Bender, G.V. Dunne, and P.N. Meisenger, Phys. Lett. A 252, 272 (1999); C.M. Bender and G.V. Dunne, J. Math. Phys. 40, 4616 (1999); G.A. Mezincescu, J. Phys. A 33, 4911 (2000); E. Delabaere and D.T. Trinh, ibid. 33, 8771 (2000); B. Bagchi and R. Roychoudhury, ibid. 33, L1 (2000); A. Khare and B.P. Mandal, Phys. Lett. A 272, 53 (2000); B. Bagchi, F. Cannata, and C. Quesne, ibid. 269, 79 (2000); M. Znojil, F. Cannata, B. Bagchi, and R. Roychoudhury, Phys. Lett. B 483, 284 (2000); M. Znojil and M. Tater, J. Phys. A 34, 1793 (2001); C.M. Bender, G.V. Dunne, P.N. Meisenger, and M. Şimşek, Phys. Lett. A 281, 311 (2001); F. Cannata, M. Ioffe, R. Roychoudhury, and P. Roy, ibid. 281, 305 (2001); Z. Ahmed, ibid. 282, 343 (2001); 284, 231 (2001); 290, 19 (2001); P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, 5679 (2001); K.C. Shin, J. Math. Phys. 42, 2513 (2001); "On the reality of the eigenvalues for a class of PT-symmetric oscillators," LANL archives, math-ph/0201013. See also S.M. Klishevich and M. Plyushchay, Nucl. Phys. B 628, 217 (2002).
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