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

SIGMA 6 (2010), 009, 8 pages      arXiv:1001.3656      https://doi.org/10.3842/SIGMA.2010.009
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

PT Symmetric Schrödinger Operators: Reality of the Perturbed Eigenvalues

Emanuela Caliceti a, Francesco Cannata b and Sandro Graffi a
a) Dipartimento di Matematica, Università di Bologna, and INFN, Bologna, Italy
b) INFN, Via Irnerio 46, 40126 Bologna, Italy

Received November 03, 2009, in final form January 14, 2010; Published online January 20, 2010

Abstract
We prove the reality of the perturbed eigenvalues of some PT symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one-dimensional x2(ix)ε for −1<ε<0.

Key words: PT symmetry; real spectra; perturbation theory.

pdf (221 kb)   ps (151 kb)   tex (12 kb)

References

  1. Caliceti E., Graffi S., Sjöstrand J., Spectra of PT-symmetric operators and perturbation theory, J. Phys. A: Math. Gen. 38 (2005), 185-193, math-ph/0407052.
  2. Caliceti E., Graffi S., Sjöstrand J., PT symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with a real discrete spectrum, J. Phys. A: Math. Theor. 40 (2007), 10155-10170, arXiv:0705.4218.
  3. Caliceti E., Cannata F., Graffi S., Perturbation theory of PT symmetric Hamiltonians, J. Phys. A: Math. Gen. 39 (2006), 10019-10027, math-ph/0607039.
  4. Rotter I., Real eigenvalues in non-Hermitian quantum physics, arXiv:0909.1232.
  5. Guo A., Salamo G.J., Duchesne D., Morandotti R., Volatier-Ravat M., Aimez V., Siviloglou G.A., Chistodoulides D.N., Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett. 103 (2009), 093902, 4 pages.
  6. Seyranian A.P., Kirillov O.N., Mailybaev A.A., Coupling of eigenvalues of complex matrices at diabolic and exceptional points, J. Phys. A: Math. Gen. 38 (2005), 1732-1740, math-ph/0411024.
  7. Kirillov O.N., Mailybaev A.A., Seyranian A.P., Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation, J. Phys. A: Math. Gen. 38 (2005), 5531-5546, math-ph/0411006.
  8. Albeverio S., Motovilov A.K., Shkalikov A.A., Bounds on variation of spectral subspaces under J-self-adjoint perturbations, Integral Equations Operator Theory 64 (2009), 455-486, arXiv:0808.2783.
  9. Albeverio S., Motovilov A.K., Tretter C., Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator, arXiv:0909.1211.
  10. Caliceti E., Cannata F., Graffi S., An analytic family of PT-symmetric Hamiltonians with real eigenvalues, J. Phys. A: Math. Theor. 41 (2008), 244008, 6 pages.
  11. Nanayakkara A., Real eigenspectra in non-Hermitian multidimensional Hamiltonians, Phys. Lett. A 304 (2002), 67-72.
  12. Nanayakkara A., Classical motion of complex 2-D non-Hermitian Hamiltonian systems, Czechoslovak J. Phys. 54 (2004), 101-107.
  13. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
  14. Dorey P., Dunning C., Tateo R., From PT-symmetric quantum mechanics to conformal field theory, Pramana J. Phys. 73 (2009), 217-239, arXiv:0906.1130.
  15. Dorey P., Dunning C., Lishman A., Tateo R., PT symmetry breaking and exceptional points for a class of inhomogeneous complex potentials, J. Phys. A: Math. Theor. 42 (2009), 465302, 41 pages. arXiv:0907.3673.
  16. Bender C.M., Besseghir K., Jones H.F., Yin X., Small-ε behavior of the non-Hermitian PT-symmetric Hamiltonian H=p2+x2(ix)ε, J. Phys. A: Math. Theor. 42 (2009), 355301, 10 pages, arXiv:0906.1291.
  17. Dorey P., Dunning C., Tateo R., Supersymmetry and the spontaneous breakdown of PT symmetry, J. Phys. A: Math. Gen. 34 (2001), L391-L400, hep-th/0104119.
  18. 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.
  19. Kato T., Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin - New York, 1976.
  20. Reed M., Simon B., Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York - London, 1978.
  21. Vock E., Hunziker W., Stability of Schrödinger eigenvalue problems, Comm. Math. Phys. 83 (1982), 281-302.
  22. Bender C.M., Holm D.D., Hook D.W., Complexified dynamical systems, J. Phys. A: Math. Theor. 40 (2007), F793-F804, arXiv:0705.3893.

Previous article   Next article   Contents of Volume 6 (2010)