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In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary.
A. Pais and G. E. Uhlenbeck, Phys. Rev. 79
1950
Earlier work this paper cites.
T. D. Lee, Phys. Rev. 95
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J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields
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The presence of nonstationary wave functions in Jordan-Block theories has been discussed in A. P. Seyranian and A. A. Mailybaev, Multiparameter stability theory with mechanical applications
1971
Earlier work this paper cites.
F. G. Scholtz, H. B. Geyer, and F. J. W. Hahne, Ann. Phys. 213
1992
Earlier work this paper cites.
C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80
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C. M. Bender, S. Boettcher, P. N. Meisinger, and Q. Wang, Phys. Lett. A 302
2002
Earlier work this paper cites.
A. Mostafazadeh, J. Math. Phys. 43
2003
Cited alongside, same era.
C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89
2004
Cited alongside, same era.
P. Dorey, C. Dunning and R. Tateo, J. Phys. A 34
2004
Cited alongside, same era.
M. S. Swanson, J. Math. Phys. 45
2004
Cited alongside, same era.
C. M. Bender, S. F. Brandt, J.-H. Chen, and Q. Wang, Phys. Rev. D 71
2005
Cited alongside, same era.
C. M. Bender, H. F. Jones, and R. J. Rivers, Phys. Lett. B625
2005
Cited alongside, same era.
P. D. Mannheim and A. Davidson, Phys. Rev. A 71
2005
Some earlier studies of 𝒫 𝒯 \mathcal{P}\mathcal{T} -symmetric Hamiltonians of Jordan-block form may be found in A. Mostafazadeh, J. Math. Phys. 43
2005
Later among the works it cites.
A. A. Andrianov, Ann. Phys. (N.Y.) 140
2006
Later among the works it cites.
P. D. Mannheim, Prog. Part. Nucl. Phys. 56
2006
Later among the works it cites.
C. M. Bender, Contemp. Phys. 46
2007
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P. Dorey, C. Dunning, and R. Tateo, J. Phys. A: Math. Gen. 40
2007
Later among the works it cites.
P. D. Mannheim, Found. Phys. 37
2007
Later among the works it cites.
C. M. Bender and P. D. Mannheim, Phys. Rev. Lett. 100
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Cited alongside, same era.
C. M. Bender and H. F. Jones, arXiv:0709.3605v1 [hep-th]
Cited in the paper.
C. M. Bender and D. W. Hook, arXiv:0802.2910v1 [hep-th]
Cited in the paper.
D. N. Christodoulides, private communication
Cited in the paper.
Jordan showed that via a sequence of similarity transformations any N N -dimensional square matrix can be brought to a form where it is either diagonal or in triangular Jordan-block form in which all of the elements on one side of the diagonal are zero. Since the nonzero elements on the other side of the diagonal do not contribute to the secular equation, the N N elements on the diagonal of a Jordan-block form matrix are then the N N eigenvalues of the matrix. When all of the eigenvalues are real, a Jordan-block matrix is an example of a non-Hermitian matrix having eigenvalues that are all real. (Hermiticity is only a sufficient condition for obtaining real eigenvalues, not a necessary one. The eigenvalues of a Hermitian matrix are real, but there is nothing that requires the eigenvalues of a non-Hermitian matrix to be complex. Non-Hermitian matrices may have real eigenvalues.) Because Jordan-block matrices cannot be brought to a diagonal form by any further similarity transform, they possess fewer than N N eigenvectors and the eigenvectors of a Jordan block matrix do not form a complete basis
Cited in the paper.
P. D. Mannheim and A. Davidson, arXiv: hep-th/0001115
Cited in the paper.
2008
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