Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 007, 24 pages      arXiv:0901.2916      https://doi.org/10.3842/SIGMA.2009.007
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

On the Spectrum of a Discrete Non-Hermitian Quantum System

Ebru Ergun
Department of Physics, Ankara University, 06100 Tandogan, Ankara, Turkey

Received October 28, 2008, in final form January 13, 2009; Published online January 19, 2009

Abstract
In this paper, we develop spectral analysis of a discrete non-Hermitian quantum system that is a discrete counterpart of some continuous quantum systems on a complex contour. In particular, simple conditions for discreteness of the spectrum are established.

Key words: difference operator; non-Hermiticity; spectrum; eigenvalue; eigenvector; completely continuous operator.

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References

  1. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947-1018, hep-th/0703096.
  2. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT-symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
  3. Dorey P., Dunning C., Tateo T., Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics, J. Phys. A: Math. Gen. 34 (2001), 5679-5704, hep-th/0103051.
  4. Shin K.C., On the reality of the eigenvalues for a class of PT-symmetric oscillators, Comm. Math. Phys. 229 (2002), 543-564, math-ph/0201013.
  5. Mostafazadeh A., Pseudo-Hermitian description of PT-symmetric systems defined on a complex contour, J. Phys. A: Math. Gen. 38 (2005), 3213-3234, quant-ph/0410012.
  6. Kelley W.G., Peterson A.C., Difference equations. An introduction with applications, Academic Press, Inc., Boston, MA, 1991.
  7. Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, Vol. 72, American Mathematical Society, Providence, RI, 2000.
  8. Bender C.M., Meisinger P.N., Wang Q., Finite dimensional PT-symmetric Hamiltonians, J. Phys. A: Math. Gen. 36 (2003), 6791-6797, quant-ph/0303174.
  9. Weigret S., How to test for digonalizability: the discretized PT-invariant square-well potential, Czechoslovak J. Phys. 55 (2005), 1183-1186.
  10. Znojil M., Matching method and exact solvability of discrete PT-symmetric square wells, J. Phys. A: Math. Gen. 39 (2006), 10247-10261, quant-ph/0605209.
  11. Znojil M., Maximal couplings in PT-symmetric chain models with the real spectrum of energies, J. Phys. A: Math. Theor. 40 (2007), 4863-4875, math-ph/0703070.
  12. Znojil M., Tridiagonal PT-symmetric N by N Hamiltonians and fine-tuning of their obsevability domains in the strongly non-Hermitian regime, J. Phys. A: Math. Theor. 40 (2007), 13131-13148, arXiv:0709.1569.
  13. Jones H.F., Scattering from localized non-Hermitian potentials, Phys. Rev. D 76 (2007), 125003, 5 pages, arXiv:0707.3031.
  14. Znojil M., Scattering theory with localized non-Hermiticities, Phys. Rev. D 78 (2008), 025026, 10 pages, arXiv:0805.2800.
  15. Ergun E., On the reality of the spectrum of a non-Hermitian discrete Hamiltonian, Rep. Math. Phys. 63 (2009), 75-93.
  16. Akhiezer N.I., Glazman I.M., Theory of linear operators in Hilbert space, Vol. 1, Ungar, New York, 1961.
  17. Lusternik L.A., Sobolev V.J., Elements of functional analysis, H. Ward Crowley Frederick Ungar Publishing Co., New York, 1961.

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