You have just completed your registration at OpenAire.
Before you can login to the site, you will need to activate your account.
An e-mail will be sent to you with the proper instructions.
Important!
Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version
of the site upon release.
We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space $\Lc^2(\Bbb R^2)$, but instead only D-quasi bases. As recently proved by one of us (FB), this is sufficient to deduce several interesting consequences.
[1] C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80, 5243-5246 (1998).
[2] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou and D.N. Christodoulides, Observation of PT -Symmetry Breaking in Complex Optical Potentials, Phys. Rev. Lett. 103, 093902 (2009).
[3] C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of parity-time symmetry in optics, Nat. Phys. 6, 1515 (2010).
[4] K. F. Zhao, M. Schaden and Z. Wu, Enhanced magnetic resonance signal of spin-polarized Rb atoms near surfaces of coated cells, Phys. Rev. A, 81, 042903 (2010).
[5] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis and T. Kottos, Experimental study of active LRC circuits with PT symmetries, Phys. Rev. A, 84, 040101, (2011).
[6] C. M. Bender, Making Sense of Non-Hermitian Hamiltonians, Rep. Progr. Phys., 70, 947-1018 (2007).
[9] J. DieudonnÂ´e, Quasi-hermitian operators, Proceedings of the International Symposium on Linear Spaces, Jerusalem 1960, Pergamon, Oxford, 115-122 (1961).
[10] F. G. Scholtz, H. B. Geyer, F.J.W. Hahne, Quasi-hermitian operators in quantum mechanics and the variational principle, Ann. Phys. 213, 74-101 (1992).
[20] F. Bagarello, Deformed canonical (anti-)commutation relations and non hermitian hamiltonians, in Non-selfadjoint operators in quantum physics: Mathematical aspects, F. Bagarello, J. P. Gazeau, F. Szafraniek and M. Znojil Eds, J. Wiley and Sons, in preparation