virtual book
on
CRYPTOHERMITICITY

a concise review of my recent results
on quantum mechanics with
Hermiticity in disguise:

A. THEORY AND MATH
- a. CONCEPTS:
- b. IDEAS:
- c. PROPOSALS:
B. PHYS AND SOLUTIONS
- d. RELATIVISTIC MODELS:
- e. CHAIN MODELS:
- f. TOBOGGANS:
C. COMPACT REVIEWS
- g. THEORY:
- h. MODELS:
- ch. APPLICATIONS:
a. theoretical concepts
1. TIME EVOLUTION AND CONSERVATION LAWS
  jump to item 52 below
2. CONDITIONAL OBSERVABILITY AND QUANTUM CATASTROPHES
  jump to item 15 below, jump to item 10 below and jump to item 5 below
3. THE AMBIGUITY OF METRIC
  jump to item 9 below, jump to item aa below
4. THREE-HILBERT-SPACE PICTURE OF QUANTUM SYSTEMS
  jump to item 7 below
return upwards

b. ideas, explanations
1. THE FRAGILITY OF THE REALITY OF THE SPECTRA
  jump to item 48 below
2. PT-ASYMMETRIC MODELS
  jump to item 43 below, jump to item 42 below
3. ANOMALOUS COUPLINGS OF CHANNELS
  jump to item 28 below
4. MATRIX NON-SQUARE HAMILTONIANS
  jump to item 25 below
return upwards

c. proposals and conjectures
1. NON-HERMITIAN SPIN, COMPLEX MASS
  jump to item 32 below, jump to item 13 below
2. NON-HERMITIAN METRIC, REINTERPRETATION OF GHOSTS
  jump to item 29 below, jump to item a53 below
3. NON-HERMITIAN BOUNDARY CONDITIONS
  jump to item 27 below
4. DISCRETE COORDINATES
  jump to item 26 below
return upwards

d. relativistic models
1. ENERGY-DEPENDENT KLEIN-GORDON
  jump to item 47 below
2. SUPERSYMMETRIC KLEIN-GORDON
  jump to item 45 below
3. KLEIN-GORDON WITH POINT INTERACTIONS
  jump to item 33 below
4. PROCA FIELD
  jump to item 17 below
return upwards

e. chain models
1. TWO BY TWO
  jump to item 23 below, jump to item 18 below
2. FOUR BY FOUR
  jump to item 20 below
3. N BY N
  jump to item 19 below
4. ELEVEN BY ELEVEN
  jump to item 8 below
return upwards

f. toboggans
1. MULTISHEETED WAVE FUNCTIONS
  jump to item 37 below
2. TOPOLOGY AND SCATTERING
  jump to item 22 below
3. MONODROMIES FOR TWO BRANCH POINTS
  jump to item 11 below
4. RECTIFICATIONS, METRIC OPERATORS, STURMIANS
  jump to item 3 below, jump to item bb below
return upwards

g. state of art
1. CLASSICAL LIMIT
  jump to item 30 below
2. REGULARIZATIONS, REDUCTIONS, DECOMPLEXIFICATIONS
  jump to item 50 below, jump to item 38 below jump to item 21 below
3. REVIEWS OF THEORETICAL EXPERIMENTS
  jump to item 51 below, jump to item 24 below, jump to item 6 below
4. EXCEPTIONAL-POINT MANIFOLDS
  jump to item 12 below
return upwards

h. models of dynamics
1. ENERGY-DEPENDENT HAMILTONIANS
  jump to item 49 below
2. PERIODIC BOUNDARY CONDITIONS
  jump to item 46 below, jump to item 35 below
3. POINT AND PIECEWISE CONSTANT INTERACTIONS
  jump to item 40 below, jump to item 39 below, jump to item 34 below
4. NEAREST-NEIGHBOR INTERACTIONS
  jump to item 14 below
return upwards

ch. new horizons in physics
1. ADVANCED APPLICATIONS IN MAGNETOHYDRODYNAMICS
  jump to item 41 below, jump to item 16 below
2. SUSY MODELS
  jump to item 36 below, jump to item 31 below
3. ADVANCED MANYBODY MODELS
  jump to item 4 below
4. CONSISTENT PICTURE OF SCATTERING
  jump to item 1 below, jump to item 2 below
return upwards

THE END
- APPENDIX.
  TIME-ORDERED LIST OF REFERENCES
1. Miloslav Znojil,
  Discrete PT-symmetric models of scattering
  J. Phys. A: Math. Theor. 41 (2008) 292002
  (arXiv:0806.2019v1 [quant-ph] 12 Jun 2008)
2. Miloslav Znojil,
  Scattering theory with localized non-Hermiticities
  Phys. Rev. D 78 (2008) 025026
  (arXiv:0805.2800v1 [hep-th] 19 May 2008)
3. Miloslav Znojil,
  Identification of observables in quantum toboggans
  J. Phys. A: Math. Theor. 41 (2008) 215304.
  (arXiv:0803.0403v1 [quant-ph] 4 Mar 2008
  and 0803.0403v2 [quant-ph] 21 Apr 2008)
4. Andreas Fring and Miloslav Znojil,
  PT-symmetric deformations of Calogero models.
  J. Phys. A: Math. Theor. 41 (2008) 194010.
  electronic and doi:10.1088/1751-8113/41/19/194010
  (arXiv: 0802.0624v1 [quant-ph] 5 Feb 2008)
5. Miloslav Znojil,
  Conditional observability versus self-duality in a schematic model.
  J. Phys. A: Math. Theor. 41 (2008) 304027,
  doi:10.1088/1751-8113/41/30/304027
  (arXiv: 0710.0457v2 [quant-ph] 17 Feb 2008)
6. Miloslav Znojil,
  Quantum toboggans: models exhibiting a multisheeted PT symmetry
  J. Phys.: Conference Series, to appear
  {IOP, ISSN 1742-6588 (Print), ISSN 1742-6596 (Online)}
  [written version of the talk in Valladolid (QTS5)]
  (arXiv:0710.1485v1 [quant-ph] 8 Oct 2007)
7. Miloslav Znojil,
  Quantum knots.
  Phys. Lett. A 372/20 pp 3591-3596 (2008)
  doi: 10.1016/j.physleta.2008.02.016
  (arXiv: 0802.1318v1 [quant-ph] 10 Feb 2008)
8. Miloslav Znojil,
  Horizons of stability.
  J. Phys. A: Math. Theor. 41 (2008) 244027
  electronic and doi:10.1088/1751-8113/41/24/244027
  (arXiv: 0801.0359 [math-ph] 2 Jan 2008)
9. Miloslav Znojil,
  On the role of the normalization factors $\kappa_n$
  and of the pseudo-metric P in crypto-Hermitian quantum models.
  SYMMETRY, INTEGRABILITY and GEOMETRY: METHODS and APPLICATIONS
  SIGMA 4 (2008), 001, 9 pages;
  (arXiv: 0710.4432v3 [math-ph] 2 Jan 2008)
10. Jun-Hua Chen, Edita Pelantova and Miloslav Znojil,
  Classification of the conditionally observable spectra exhibiting central symmetry.
  Phys. Lett. A, Volume 372, Issue 12, 17 March 2008, Pages 1986-1989;
  available in electronic form: doi:10.1016/j.physleta.2007.11.015
  (arXiv: 0711.3947v1 [math-ph] 26 Nov 2007)
11. Miloslav Znojil,
  Quantum toboggans with two branch points
  Phys. Lett. A 372, Issue 5, 28 January 2008, Pages 584-590
  http://dx.doi.org/10.1016/j.physleta.2007.07.072
  (arXiv:0708.0087v1 [quant-ph] 1 Aug 2007)
12. Miloslav Znojil,
  Tridiagonal PT-symmetric N by N Hamiltonians
  and a fine-tuning of their observability domains in the strongly non-Hermitian regime.
  J. Phys. A: Math. Theor. 40 (2007) 13131-13148
  (arXiv: 0709.1569v1 [math-ph] 11 Sep 2007)
  doi:10.1088/1751-8113/40/43/020
13. Miloslav Znojil,
  Mass-sign duality of cubic oscillators
  Advanced Studies in Theoretical Physics, Vol. 1, 2007, no. 8, 405 - 408,
  ISSN 1313-1311 (math-ph/0612073 or pdf)
14. Miloslav Znojil,
  PT-symmetric quantum chain models
  Acta Polytechnica 47 (2007) 9 - 14
  ISSN 1210-2709
  (presented also as a talk during the third DI microconference AAMP II )
15. Miloslav Znojil,
  Conditional observability
  Phys. Lett. B 650 (2007) 440 - 446
  (arXiv:0704.3812v1 [hep-th] 28 Apr 2007)
16. Miloslav Znojil and Uwe Guenther,
  Dynamics of charged fluids and 1/L perturbation expansions
  J. Phys. A: Math. Theor. 40 (2007) 7375-7388.
  (math-ph/0610055v2)
17. Jaroslav Smejkal, Vit Jakubsky and Miloslav Znojil,
  Relativistic vector bosons and PT-symmetry
  Journal of Physical Studies 11 (2007) 45 - 54.
  ISSN 1027-4642 (hep-th/0611287)
18. Miloslav Znojil and Hendrik B. Geyer,
  Construction of a unique metric in quasi-Hermitian quantum mechanics (erratum)
  Phys. Lett. B 649 (2007) 494
19. Miloslav Znojil,
  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).
20. Miloslav Znojil,
  Determination of the domain of the admissible matrix elements in the four-dimensional PT-symmetric anharmonic model
  Phys. Lett. A 367, Issues 4-5, 30 July 2007, Pages 300-306 (available online)
  (quant-ph/0703168).
21. Miloslav Znojil,
  A return to observability near exceptional points in a schematic PT-symmetric model
  Phys. Lett. B 647 (2007) 225 - 230
  (quant-ph/0701232).
22. Miloslav Znojil,
  Spiked potentials and quantum toboggans
  J. Phys. A: Math. Gen. 39 (2006) 13325-13336
  (quant-ph/0606166v2).
23. Miloslav Znojil and Hendrik B. Geyer,
  Construction of a unique metric in quasi-Hermitian quantum mechanics:
  non-existence of the charge operator in a 2 x 2 matrix model
  Phys. Lett. B 640 (2006) 52 - 56
  (quant-ph/0607104)
24. Miloslav Znojil,
  On a few new quantization recipes using PT-symmetry
  Czechosl. J. Phys. 56 (2006) 977 - 984
  (September, special issue)
25. Miloslav Znojil,
  Quasi-exact minus-quartic oscillators in strong-core regime
  Phys. Lett. A 359 (2006) 21 - 25
  (quant-ph/0602231)
26. Miloslav Znojil,
  Matching method and exact solvability of discrete PT-symmetric square wells
  J. Phys. A: Math. Gen. 39 (2006) 10247 - 10261
  (quant-ph/0605209)
27. David Krejčiřík, Hynek Bíla and Miloslav Znojil,
  Closed formula for the metric in the Hilbert space of a PT-symmetric model
  J. Phys. A: Math. Gen. 39 (2006) 10143 - 10153
  (math-ph/0604055)
28. Miloslav Znojil,
  Exactly solvable models with PT-symmetry and with an asymmetric coupling of channels
  J. Phys. A: Math. Gen. 39 (2006) 4047 - 4061
  (quant-ph/0511194)
29. Miloslav Znojil,
  Strengthened PT-symmetry with P $\neq$ P$^\dagger$
  Phys. Lett. A 353 (2006) 463 - 468
  (quant-ph/0601048)
30. H. Bila, M. Tater and M. Znojil,
  'Comment on ``Comparison of quantal and classical behavior of PT-symmetric systems at avoided crossings"
  [A. Nananyakkara, Phys. Lett. A 334 (2005) 144]'
  Phys. Lett. A 351 (2006) 452 - 456
31. B. Bagchi, H. Bila, V. Jakubsky, S. Mallik, C. Quesne and M. Znojil,
  PT-symmetric supersymmetry in a solvable short-range model
  Int. J. Mod. Phys. A [Particles and Fields; Gravitation; Cosmology; Nuclear Physics] 21 (2006) 2173-2190
  (quant-ph/0503035)
32. Miloslav Znojil,
  Coupled-channel version of PT-symmetric square well
  J. Phys. A: Math. Gen. 39 (2006) 441 - 455.
  (quant-ph/0511085)
33. Miloslav Znojil,
  Solvable relativistic quantum dots with vibrational spectra.
  (quant-ph/0506017) Czech. J. Phys. 55 (2005) 1187 - 1192
34. C. Quesne, B. Bagchi, S. Mallik, H. Bila, V. Jakubsky and M. Znojil,
  PT-supersymmetric partner of a short-range square well.
  Czech. J. Phys. 55 (2005) 1161 - 1166
35. Vit Jakubský and Miloslav Znojil,
  An explicitly solvable model of the spontaneous PT-symmetry breaking
  (quant-ph/0507132) Czech. J. Phys. 55 (2005) 1113 - 1116
36. H. Bila, V. Jakubsky, M. Znojil, B. Bagchi, S. Mallik and C. Quesne,
  Weakly non-Hermitian square well
  Czech. J. Phys. 55 (2005) 1075 - 1076
37. Miloslav Znojil,
  PT-symmetric quantum toboggans
  DOI information: doi: http://dx.doi.org/10.1016/j.physleta.2005.05.029
  (quant-ph/0502041) Phys. Lett. A 342 (2005) 36-47.
38. Miloslav Znojil,
  Perturbation method for non-square Hamiltonians and its application to polynomial oscillators
  (math-ph/0503011v2), Phys. Lett. A 341 (2005) 67 - 80.
39. M. Znojil,
  Solvable PT-symmetric model with a tunable interspersion of non-merging levels
  (quant-ph/0410196), J. Math. Phys. 46 (2005) 062109
40. Miloslav Znojil and Vít Jakubský,
  Solvability and PT-symmetry in a double-well model with point interactions
  (electronic: quant-ph/0503235, doi: click or the journal's temporarily free access),
  J. Phys. A: Math. Gen. 38 (2005) 5041-5056.
41. Uwe Guenther, Frank Stefani and Miloslav Znojil,
  MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator
  (math-ph/0501069, doi:10.1063/1.1915293), J. Math. Phys. 46, 063504 (2005)
42. B. Bagchi, A. Banerjee, E. Caliceti, F. Cannata, H. B. Geyer, C. Quesne and M. Znojil,
  CPT-conserving Hamiltonians and their nonlinear supersymmetrization using differential charge-operators C
  Int. J. Mod. Phys. A 20 (2005) 7107 - 7128.
  (hep-th/0412211),
43. Emanuela Caliceti, Francesco Cannata, Miloslav Znojil and Alberto Ventura,
  Construction of PT-asymmetric non-Hermitian Hamiltonians with CPT-symmetry.
  Phys. Lett. A 335 (2005) 26-30
  DOI information: doi: http://dx.doi.org/10.1016/j.physleta.2004.12.004
  (math-ph/0406031)
44. Miloslav Znojil
  The method of Hill determinants in PT-symmetric quantum mechanics
  in ``Recent Trends in Exponential Asymptotics" , ed. Y. Takei,
  RIMS Kokyuroku series vol. 1424 (2005) pp. 240 - 253
  (math-ph/0410009, proceedings of the conference held in RIMS, Kyoto, June 28 - July 2, 2004)
45. M. Znojil,
  Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation.
  J. Phys. A: Math. Gen. 37 (2004) 9557-9571
  (hep-th/0408232)
46. V. Jakubsky and M. Znojil,
  Periodic square-well potential and spontaneous breakdown of PT-symmetry.
  Czech. J. Phys. 54 (2004) 1101 - 1106
  (quant-ph/0408189)
47. M. Znojil, H. Bila and V. Jakubsky,
  Pseudo-Hermitian approach to energy-dependent Klein-Gordon models.
  Czech. J. Phys. 54 (2004) 1143 - 1148
  (quant-ph/0408141).
48. M. Znojil,
  Fragile PT-symmetry in a solvable model.
  J. Math. Phys. 45 (2004) 4418 - 30
  (math-ph/0403033),
49. M. Znojil,
  Linear representation of energy-dependent Hamiltonians
  Phys. Lett. A 326 (2004) 70 - 76
  (quant-ph/0403223).
50. M. Znojil,
  PT-symmetric regularizations in supersymmetric quantum mechanics
  J. Phys. A: Math. Gen. 37 (2004) 10209-10222
  (hep-th/0404145).
51. Miloslav Znojil,
  Experiments in PT-symmetric quantum mechanics
  Czechoslovak J. Phys. 54 (2004) 151 -- 156
  (quant-ph/0309100v2).
52. M. Znojil,
  Conservation of pseudo-norm in PT symmetric quantum mechanics
  Rendiconti del Circ. Mat. di Palermo, Ser. II, Suppl. 72 (2004), pp. 211 - 218
  (math-ph/0104012).
53. M. Znojil,
  PT-symmetry, ghosts, SUSY and Klein-Gordon equation.
  (hep-th/0408081), invited talk, appeared in "Symmetry Methods in Physics", proceedings of the XI-th International Conference Prague, June 21 - 24, 2004
  Ed. by . Burdik, O. Navratil and S. Posta
  published (on CD ROM, plus booklet of abstracts) by
  JINR, Dubna, 2004
54. M. Znojil
  New types of solvability in PT symmetric quantum theory,
  in "Superintegrability in Classical and Quantum Systems",
  Ed. P.Tempesta, P. Winternitz, J. Harnad, W. Miller, Jr., G. Pogosyan and M. Rodriguez,
  CRM Proceedings et Lecture Notes 37, AMS, Providence, 2004, pp. 333-347
  (math-ph/0501058, invited talk for
  Workshop on Superintegrability in Classical and Quantum Systems
  September 16-21, 2002, Montreal, Canada).
55. Miloslav Znojil and Denis Yanovich,
  New type of exact solvability and of a hidden nonlinear dynamical symmetry in anharmonic oscillators
  in The Proceedings of Fifth International Conference
  SYMMETRY IN NONLINEAR MATHEMATICAL PHYSICS , June 23-29, 2003,
  (Proc. Instit. Math. NAS of Ukraine, vol. 50, part II, ISBN 966-02-3226-8),
  Ed. A. G. Nikitin, V. M. Boyko, R. O. Popovych and I. A. Yehorchenko,
  Institute of Mathematics, Kyiv, 2004, pp. 1010 - 1017
  (math-ph/0309047).
56. a44: Hill in Kyoto
  in proceedings - see arXiv
57. a53: ghosts in Prague
  in proceedings - see arXiv
58. a54: anomalous solvability in Montreal
  in proceedings - see arXiv
59. a55: Yanovitch
  in proceedings - see arXiv
60. aa: the construction of all the metrics Theta for a given H
  preprint, submitted - see arXiv
61. bb: all the Sturmian metrics Theta for a given H and W
  preprint, submitted - see arXiv
return upwards

date: August 18, 2008

virtual book on CRYPTOHERMITICITY

a concise review of my recent results on quantum mechanics with Hermiticity in disguise:

contents:

contents:

A. THEORY AND MATH

a. CONCEPTS:

b. IDEAS:

c. PROPOSALS:

B. PHYS AND SOLUTIONS

d. RELATIVISTIC MODELS:

e. CHAIN MODELS:

f. TOBOGGANS:

C. COMPACT REVIEWS

g. THEORY:

h. MODELS:

ch. APPLICATIONS:

a. theoretical concepts

TIME EVOLUTION AND CONSERVATION LAWS

CONDITIONAL OBSERVABILITY AND QUANTUM CATASTROPHES

THE AMBIGUITY OF METRIC

THREE-HILBERT-SPACE PICTURE OF QUANTUM SYSTEMS

b. ideas, explanations

THE FRAGILITY OF THE REALITY OF THE SPECTRA

PT-ASYMMETRIC MODELS

ANOMALOUS COUPLINGS OF CHANNELS

MATRIX NON-SQUARE HAMILTONIANS

c. proposals and conjectures

NON-HERMITIAN SPIN, COMPLEX MASS

NON-HERMITIAN METRIC, REINTERPRETATION OF GHOSTS

NON-HERMITIAN BOUNDARY CONDITIONS

DISCRETE COORDINATES

d. relativistic models

ENERGY-DEPENDENT KLEIN-GORDON

SUPERSYMMETRIC KLEIN-GORDON

KLEIN-GORDON WITH POINT INTERACTIONS

PROCA FIELD

e. chain models

TWO BY TWO

FOUR BY FOUR

N BY N

ELEVEN BY ELEVEN

f. toboggans

MULTISHEETED WAVE FUNCTIONS

TOPOLOGY AND SCATTERING

MONODROMIES FOR TWO BRANCH POINTS

RECTIFICATIONS, METRIC OPERATORS, STURMIANS

g. state of art

CLASSICAL LIMIT

REGULARIZATIONS, REDUCTIONS, DECOMPLEXIFICATIONS

REVIEWS OF THEORETICAL EXPERIMENTS

EXCEPTIONAL-POINT MANIFOLDS

h. models of dynamics

ENERGY-DEPENDENT HAMILTONIANS

PERIODIC BOUNDARY CONDITIONS

POINT AND PIECEWISE CONSTANT INTERACTIONS

NEAREST-NEIGHBOR INTERACTIONS

ch. new horizons in physics

ADVANCED APPLICATIONS IN MAGNETOHYDRODYNAMICS

SUSY MODELS

ADVANCED MANYBODY MODELS

CONSISTENT PICTURE OF SCATTERING

THE END

APPENDIX.

TIME-ORDERED LIST OF REFERENCES

a44: Hill in Kyoto

a53: ghosts in Prague

a54: anomalous solvability in Montreal

a55: Yanovitch

aa: the construction of all the metrics Theta for a given H

bb: all the Sturmian metrics Theta for a given H and W

virtual book
on
CRYPTOHERMITICITY

a concise review of my recent results
on quantum mechanics with
Hermiticity in disguise: