# Three Solvable Matrix Models of a Quantum Catastrophe

Article

First Online:

- 123 Downloads
- 11 Citations

## Abstract

Three classes of finite-dimensional models of quantum systems exhibiting spectral degeneracies called quantum catastrophes are described in detail. Computer-assisted symbolic manipulation techniques are shown unexpectedly efficient for the purpose.

## Keywords

Quantum theory PT symmetry Finite-dimensional non-Hermitian Hamiltonians Exceptional-point localization Quantum theory of catastrophes Methods of computer algebra## References

- 1.Dorey, P., Dunning, C., Tateo, R.: Spectral equivalences, Bethe Ansatz equations, and reality properties in 𝓟𝓣-symmetric quantum mechanics. J. Phys. A: Math. Gen.
**34**, 5679–5704 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 2.Bender, C.M.: Making sense of non-hermitian Hamiltonians. Rep. Prog. Phys.
**70**, 947–1018 (2007)ADSCrossRefGoogle Scholar - 3.Dorey, P., Dunning, C., Tateo, R.: The ODE/IM correspondence. J. Phys. A: Math. Theor.
**40**, R205–R283 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 4.Mostafazadeh, A.: Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Meth. Mod. Phys.
**7**, 1191–1306 (2010)zbMATHMathSciNetGoogle Scholar - 5.Znojil, M.: Three-Hilbert-space formulation of Quantum Mechanics. Symmetry, Integrability and Geometry: Methods and Applications, vol. 5, 001, p. 19 (2009)Google Scholar
- 6.Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having pt symmetry. Phys. Rev. Lett.
**80**, 5243–5246 (1998)ADSzbMATHMathSciNetGoogle Scholar - 7.Siegl, P., Krejčiřík, D.: On the metric operator for the imaginary cubic oscillator. Phys. Rev. D
**86**, 121702(R) (2012)ADSGoogle Scholar - 8.Scholtz, F.G., Geyer, H.B., Hahne, F.J.H.: Quasi-Hermitian operators in quantum mechanics and the variational principle. Ann. Phys. (NY)
**213**, 74–101 (1992)ADSzbMATHMathSciNetGoogle Scholar - 9.Kato, T.: Perturbation theory for linear operators. Spinger, Berlin (1966)zbMATHGoogle Scholar
- 10.Znojil, M.: Quantum catastrophes: a case study. J. Phys. A: Math. Theor.
**45**, 444036 (2012)ADSMathSciNetGoogle Scholar - 11.Znojil, M.: N-site-lattice analogues of
*V*(*x*) =*ix*^{3}. Ann. Phys. (NY)**327**, 893–913 (2012)ADSzbMATHMathSciNetGoogle Scholar - 12.Jones, H.F.: Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation. Phys. Rev. D
**78**, 065032 (2008)ADSGoogle Scholar - 13.Znojil, M.: Scattering theory with localized non-Hermiticities. Phys. Rev. D
**78**, 025026 (2008)ADSMathSciNetGoogle Scholar - 14.Krejčiřík, D., Bíla, H., Znojil, M.: Closed formula for the metric in the Hilbert space of a 𝓟𝓣-symmetric model. J. Phys. A: Math. Gen.
**39**, 10143–10153 (2006)ADSzbMATHGoogle Scholar - 15.Znojil, M.: Complete set of inner products for a discrete 𝓟𝓣-symmetric square-well Hamiltonian. J. Math. Phys.
**50**, 122105 (2009)ADSMathSciNetGoogle Scholar - 16.Znojil, M., Wu, J.: A generalized family of discrete PT-symmetric square wells. Int. J. Theor. Phys.
**52**, 2152–2162 (2013)MathSciNetGoogle Scholar - 17.Znojil, M.: Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding. Ann. Phys. (NY)
**336**, 98–111 (2013)ADSzbMATHMathSciNetGoogle Scholar - 18.Znojil, M.: Maximal couplings in 𝓟𝓣-symmetric chain-models with the real spectrum of energies. J. Phys. A: Math. Theor.
**40**, 4863–4875 (2007)ADSzbMATHMathSciNetGoogle Scholar - 19.Char, B.W. et al.: Maple V Language Reference Manual. Springer, New York (1993)Google Scholar
- 20.Znojil, M.: Symbolic-manipulation constructions of Hilbert-space metrics in quantum mechanics.Lecture Notes in Computer Science
**6885**, 348–357 (2011)Google Scholar - 21.Znojil, M.: Quantum inner-product metrics via recurrent solution of Dieudonne equation. J. Phys. A: Math. Theor.
**45**, 085302 (2012)ADSMathSciNetGoogle Scholar - 22.Znojil, M.: On the role of the normalization factors
*κ*_{n}and of the pseudo-metric P in crypto-Hermitian quantum models. Symmetry, Integrability and Geometry: Methods and Applications. SIGMA**4**, 001 (2008)Google Scholar - 23.
- 24.Heiss, W.D.: The physics of exceptional points. J. Phys. A: Math. Theor.
**45**, 444016 (2012)ADSMathSciNetGoogle Scholar - 25.Znojil, M.: Quantum Big Bang without fine-tuning in a toy-model. J. Phys. Conf. Ser.
**343**, 012136 (2012)ADSGoogle Scholar - 26.Thom, R.: Structural stability and morphogenesis. An outline of a general theory of models. Benjamin, Reading (1975)Google Scholar
- 27.Arnold, V.I.: Catastrophe Theory. Springer-Verlag, Berlin (1992)Google Scholar
- 28.Langer, H., Tretter, C.: A Krein space approach to PT symmetry. Czechosl. J. Phys.
**70**, 1113–1120 (2004)ADSMathSciNetGoogle Scholar - 29.Krejčiřík, D., Siegl, P., železný, J.: On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators. Compl. Anal. Oper. Theory
**8**, 255–281 (2014). arXiv:1108.4946 - 30.Hernandez-Coronado, H., Krejčiřík, D., Siegl, P.: Perfect transmission scattering as a 𝓟𝓣-symmetric spectral problem. Phys. Lett. A
**375**, 2149–2152 (2011)ADSzbMATHMathSciNetGoogle Scholar - 31.Ambichl, P., Makris, K.G., Ge, L., Chong, Y.-D., Stone, A.D., Rotter, S.: Breaking of PT symmetry in bounded and unbounded scattering systems. Phys. Rev. X
**3**(041030), 9 (2013)Google Scholar

## Copyright information

© Springer Science+Business Media New York 2014