ON PT-SYMMETRIC HAMILTONIANS

co-authors:

B. Bagchi (Calcutta), A. Banerjee (Calcutta), H. Bila (Rez), E. Caliceti (Bologna), F. Cannata (Bologna), F. M. Fernández (La Plata), F. Gemperle (Prague), H. B. Geyer (Stellenbosch), R. Guardiola (Burjasot), U. Guenther (Rossendorf), V. Jakubsky (Rez), G. Levai (Debrecen), S. Mallik (Calcutta), O. Mustafa (Famgusta), C. Quesne (Brussels), J. Ros (Burjasot), P. Roy (Calcutta), R. Roychoudhury (Calcutta), F. Stefani (Rossendorf), M. Tater (Rez), A. Ventura (Bologna), M. Znojil (Rez)

remark:

What follows is just an electronic guide to the galaxy of my papers on PT-symmetric Quantum Mechanics.
Their time-ordered list may be found here

contents (click to)

A. PRINCIPLES

a. Origins: real energies from perturbation series
b. Concepts: numerical experiments and explanations
c. Harmonic oscillator: a solvable example

B. TECHNIQUES

d. SUSY: early applications
e. Changes of variables: a proliferation of non-numerical models
f. Square wells: insight and limitations

C. APPLICATIONS

g. Quasi-exact solvability: new horizons
h. Periodic boundary conditions: beyond single particle/dimension
ch. Klein-Gordon equation: advanced applications in physics

a. Origins: real energies from perturbation series

STRONG-COUPLING SERIES
M. F. Fernández, R. Guardiola, J. Ros and M. Znojil,
Strong-coupling expansions for the PT-symmetric oscillators V(x) = a x + b (ix)^2 + c (ix)^3.
J. Phys. A 31 (1998) 10105 - 12.
1/L EXPANSIONS
M. Znojil, F. Gemperle and O. Mustafa,
Asymptotic solvability of an imaginary cubic oscillator with spikes
(hep-th/0205181), J. Phys. A: Math. Gen. 35 (2002) 5781 - 5793.

O. Mustafa and M. Znojil,
PT symmetric pseudo-perturbation recipe: an imaginary cubic oscillator with spikes
(math-ph/0206042), J. Phys. A: Math. Gen. 35 (2002) 8929 - 8942.

M. Znojil,
Solvable simulation of a double-well problem in PT symmetric quantum mechanics
(quant-ph/0303122), J. Phys. A: Math. Gen. 36 (2003) 7639 - 7648

M. Znojil,
talk during the VIth meeting of Czech mathematical physicists
Olomouc, March 19 - 20, 2004, and text in Czech in proceedings on CD (2004)

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b. Concepts: numerical experiments and explanations

THE WEDGE-DEPENDENCE OF SPECTRA:
M. F. Fernández, R. Guardiola, J. Ros and M. Znojil,
A family of complex potentials with real spectrum.
J. Phys. A 32 (1999) 3105 - 16.
HILL-DETERMINANT METHOD - A PROOF:
M. Znojil,
Non-Hermitian matrix description of the PT-symmetric anharmonic oscillators
J. Phys. A: Math. Gen. 32 (1999) 7419 - 28.
PARTITIONING AND EFFECTIVE HAMILTONIANS:
M. Znojil,
Should PT symmetric quantum mechanics be interpreted as nonlinear?
(quant-ph/0103054v4), J. Nonlin. Math. Phys. 9, suppl. 2 (2002), 122-133

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).

M. Znojil,
A generalization of the concept of PT symmetry
(math-ph/0106021), in ``Quantum Theory and Symmetries", ed. E. Kapuscik and A. Horzela, Word Sci., Singapore,2002, pp. 626-631.
APPLICATIONS OUTSIDE QUANTUM THEORY:
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), J. Math. Phys., to appear

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c. Harmonic oscillator: a solvable example

COMPLEX SHIFT OF THE COORDINATE:
M. Znojil,
PT symmetric harmonic oscillators
Phys. Lett. A 259 (1999) 220 - 3.

B. Bagchi, C. Quesne and M. Znojil,
Generalized continuity equation and modified normalization in PT-symmetric quantum mechanics
(quant-ph/0108096) Mod. Phys. Letters A 16 (2001) 2047 - 2057
SUPERSYMMETRIC INTERPRETATION:
M. Znojil,
Non-Hermitian SUSY and singular, PT-symmetrized oscillators
(hep-th/0201056), J. Phys. A: Math. Gen. 35 (2002) 2341 - 2352.

M. Znojil,
Re-establishing supersymmetry between harmonic oscillators in D =/= 1 dimensions
(hep-th/0203252), Rendiconti del Circ. Mat. di Palermo, Serie II, Suppl. 71 (2003) 199 - 207.

M. Znojil,
``PT-symmetry and supersymmetry",
in ``GROUP 24: Physical and Mathematical Aspects of Symmetries",
Ed. J-P. Gazeau et al (IOP Publishing, Bristol, 2003). pp. 629 - 632 (hep-th/0209062).

M. Znojil,
Comment on ``Supersymmetry and Singular Potentials" by Das and Pernice [Nucl. Phys. B 561 (1999) 357]
( hep-th/0209262v2), Nucl. Phys. B 662, 554-562 (2003)

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d. Supersymmetry: early applications

SHAPE INVARIANCE:
M. Znojil,
Shape invariant potentials with PT symmetry
(quant-ph/9911116v1), J. Phys. A: Math. Gen. 33 (2000) L61 - L62.
THE VIOLATION OF HERMITICITY:
M. Znojil, F. Cannata, B. Bagchi and R. Roychoudhury,
Supersymmetry without hermiticity within PT symmetric quantum mechanics
(hep-th/0003277), Phys. Lett. B 483 (2000) 284 - 289.

M. Znojil,
PT symmetrized SUSY quantum mechanics
(hep-ph/0101038), Czechoslovak J. Phys. 51 (2001) 420 - 428.

G. Levai and M. Znojil,
The interplay of supersymmetry and ${\cal PT}$ symmetry in quantum mechanics: a case study for the Scarf II potential
(quant-ph/0206013), J. Phys. A: Math. Gen. 35 (2002) 8793 - 8804.

M. Znojil,
PT-symmetric regularizations in supersymmetric quantum mechanics
J. Phys. A: Math. Gen. 37 (2004) 10209-10222
(hep-th/0404145).
DIFFERENTIAL CHARGE OPERATORS:
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: 10.1016/j.physleta.2004.12.004
(math-ph/0406031)

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. of Modern Physics A, to appear
(hep-th/0412211),

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e. Changes of variables: a proliferation of non-numerical models

LAGUERRE POLYNOMIALS:
M. Znojil,
Exact solution for Morse oscillator in PT-symmetric quantum mechanics
Phys. Lett. A 264 (1999) 108 - 11.

M. Znojil and G. Levai,
The Coulomb - harmonic-oscillator correspondence in PT symmetric quantum mechanics
(quant-ph/0003081), Phys. Lett. A 271 (2000) 327 - 333.
JACOBI POLYNOMIALS:
M. Znojil,
PT-symmetrically regularized Eckart, Poeschl-Teller and Hulthen potentials
(quant-ph/9912027v2), J. Phys. A: Math. Gen. 33 (2000) 4561 - 4572.

M. Znojil,
Eight exactly solvable complex potentials in Bender - Boettcher quantum mechanics
Rendiconti del Circ. Mat. di Palermo, Serie II, Suppl. 66 (2001) 213 - 218.
ALL HYPERGEOMETRIC POLYNOMIALS:
G. Levai and M. Znojil,
Systematic search for PT symmetric potentials with real energy spectra
J. Phys. A: Math. Gen. 33 (2000) 7165 - 7180.

G. Lévai and M. Znojil,
Conditions for complex spectra in a class of PT symmetric potentials
(quant-ph/0110064) Mod. Phys. Letters A 16 (2001) 1973 - 1981
ANOMALIES:
M. Znojil, G. Le'vai, P. Roy and R. Roychoudhury,
Anomalous doublets of states in a PT symmetric quantum model,
(quant-ph/0104059) Phys. Lett. A 290 (2001) 249 - 254.

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f. Square wells: insight and limitations

SINGLE MATCHING POINT:
M. Znojil,
PT symmetric square well
(quant-ph/0101131), Phys. Lett. A 285 (2001) 7 - 10.

M. Znojil and G. Lévai,
Spontaneous breakdown of PT symmetry in the solvable square well model
(hep-th/0111213) Mod. Phys. Letters A 16 (2001) 2273 - 2280.

M. Znojil,
Fragile PT-symmetry in a solvable model.
J. Math. Phys. 45 (2004) 4418 - 30
(math-ph/0403033),
MORE MATCHING POINTS:
M. Znojil,
Imaginary cubic oscillator and its square-well approximations in x- and p-representation
in "Advances in Numerical Analysis (Numerical Analysis Research)",
F. Columbus, Ed. (Nova Science Publishers: Hauppauge), to appear
(quant-ph/0103115 and math-ph/0101027).

M. Znojil,
Solvable PT symmetric model with a tunable interspersion of non-merging levels
(quant-ph/0410196), J. Math. Phys., submitted

B. Bagchi, H. Bíla, V. Jakubský, S. Mallik, C. Quesne and M. Znojil,
PT symmetric supersymmetry in a solvable short-range model
(quant-ph/0503035), J. Phys. A., submitted

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g. Quasi-exact solvability: new horizons

ADDING A WEAK, COULOMBIC SPIKE:
M. Znojil,
Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable.
J. Phys. A: Math. Gen. 32 (1999) 4563 - 70.
CENTRIFUGAL SPIKE IN NEW ROLE:
M. Znojil,
Quasi-exactly solvable quartic potentials with centrifugal and Coulombic terms
(math-ph/0002036v2), J. Phys. A 33 (2000) 4203 - 11.

M. Znojil,
Spiked and PT-symmetrized decadic potentials supporting elementary N-plets of bound states
(quant-ph/0002083), J. Phys. A: Math. Gen. 33 (2000) 6825 - 6833.

M. Znojil
New types of solvability in PT symmetric quantum theory,
in "Superintegrability in Classical and Quantum Systems",
Ed. P. Tempesta et al,
Proceedings et Lecture Notes 37, AMS, Providence, 2004, pp. 333- 347
(math-ph/0501058).

M. Znojil
Quantum toboggans,
(quant-ph/0502041), submitted to J. Phys. A.

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h. Periodic boundary conditions: beyond single particle/dimension

SOLVABLE MODELS WITH MORE PARTICLES:
M. Znojil and M. Tater,
Complex Calogero model with real energies
(quant-ph/0010087), J. Phys. A: Math. Gen. 34 (2001) 1793-1803.

M. Znojil and M. Tater,
Exactly solvable three-body Calogero-type model with translucent two-body barriers
(nucl-th/0103015), Phys. Lett. A 284 (2001) 225 - 230.

Vit Jakubsky, Miloslav Znojil, Euclides Augusto Luis and Frieder Kleefeld,
Trigonometric identities, angular Schr\"{o}dinger equations and a new family of solvable models
Phys. Lett. A 334 (2005) 154 - 159
(quant-ph/0410023)
PERIODIC BOUNDARY CONDITIONS:
M. Znojil,
Solvable PT-symmetric Hamiltonians
(quant-ph/0008125), Physics of Atomic Nuclei 65 (2002) 1149 - 1151
= the English version of
Yad. Fiz. 65 (2002), 1182-1184.

V. Jakubský and M. Znojil,
Periodic square-well potential and spontaneous breakdown of PT-symmetry.
Czech. J. Phys. 54 (2004) 1101 - 1106
(quant-ph/0408189)
ANGULAR SCHROEDINGER EQUATIONS:
M. Znojil,
Poeschl-Teller paradoxes
(math-ph/0102034), J. Phys. A: Math. Gen. 34 (2001) 9585 - 9592.

M. Znojil,
PT symmetric models in more dimensions and solvable square-well versions of their angular Schroedinger equations.
(quant-ph/0304046), J. Phys. A: Math. Gen. 36 (2003) 7825 - 7838

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ch. Klein-Gordon equation: advanced applications in physics

FESHBACH - VILLARS RELATIVISTIC HAMILTONIAN:
Miloslav Znojil,
Experiments in PT-symmetric quantum mechanics
Czechoslovak J. Phys. 54 (2004) 151 -- 156
(quant-ph/0309100v2).
ENERGY-DEPENDENT INTERACTIONS:
M. Znojil,
Linear representation of energy-dependent Hamiltonians
Phys. Lett. A 326 (2004) 70 - 76
(quant-ph/0403223).

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).
SUPERSYMMETRIC RELATIVISTIC MODELS:
M. Znojil,
PT-symmetry, ghosts, SUSY and Klein-Gordon equation.
(hep-th/0408081), in XI-th International Conference "Symmetry Methods in Physics", Ed. C. Burdik et al
(CD ROM plus booklet of abstracts)
JINR, Dubna, 2004

M. Znojil,
Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation.
J. Phys. A: Math. Gen. 37 (2004) 9557-9571
(hep-th/0408232)

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THE END OF THE STORY.

APPENDIX.
SAME MATERIAL, TIME-ORDERED

M. Znojil,
The method of Hill determinants in PT-symmetric quantum mechanics
(math-ph/0410009), a review lecture for Proceedings ``Recent Trends in Exponential Asymptotics" (June 28 - July 2, 2004, RIMS, Kyoto, JAPAN), to appear in proceedings

B. Bagchi, H. Bila, V. Jakubsky, S. Mallik, C. Quesne and M. Znojil,
PT-symmetric supersymmetry in a solvable short-range model
(quant-ph/0503035), submitted to J. Phys. A

Miloslav Znojil,
Quantum toboggans
(quant-ph/0502041), submitted to J. Phys. A: Math. Gen.

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), submitted to J. Math. Phys.

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
(hep-th/0412211), submitted to Int. J. of Modern Physics A

M. Znojil,
Solvable PT-symmetric model with a tunable interspersion of non-merging levels
(quant-ph/0410196), submitted to J. Math. Phys.

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: 10.1016/j.physleta.2004.12.004
(math-ph/0406031)
M. Znojil,
Imaginary cubic oscillator and its square-well approximations in x- and p-representation
in "Advances in Numerical Analysis (Numerical Analysis Research)",
Frank Columbus, Editor (Nova Science Publishers: Hauppauge), to appear
(quant-ph/0103115 and math-ph/0101027).
M. Znojil,
Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation.
J. Phys. A: Math. Gen. 37 (2004) 9557-9571
(hep-th/0408232)
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)
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).
M. Znojil,
Fragile PT-symmetry in a solvable model.
J. Math. Phys. 45 (2004) 4418 - 30
(math-ph/0403033),
M. Znojil,
Linear representation of energy-dependent Hamiltonians
Phys. Lett. A 326 (2004) 70 - 76
(quant-ph/0403223).
M. Znojil,
PT-symmetric regularizations in supersymmetric quantum mechanics
J. Phys. A: Math. Gen. 37 (2004) 10209-10222
(hep-th/0404145).
Miloslav Znojil,
Experiments in PT-symmetric quantum mechanics
Czechoslovak J. Phys. 54 (2004) 151 -- 156
(quant-ph/0309100v2).
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).
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

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
( 15pp., invited talk for
Workshop on Superintegrability in Classical and Quantum Systems
September 16-21, 2002, Montreal, Canada)
(math-ph/0501058).

M. Znojil,
PT symmetric models in more dimensions and solvable square-well versions of their angular Schroedinger equations.
(quant-ph/0304046), J. Phys. A: Math. Gen. 36 (2003) 7825 - 7838
M. Znojil,
Solvable simulation of a double-well problem in PT symmetric quantum mechanics
(quant-ph/0303122), J. Phys. A: Math. Gen. 36 (2003) 7639 - 7648
M. Znojil,
Re-establishing supersymmetry between harmonic oscillators in D =/= 1 dimensions
(hep-th/0203252), Rendiconti del Circ. Mat. di Palermo, Serie II, Suppl. 71 (2003) 199 - 207 (vol. of proc. of 22nd Winter School ``Geometry and Physics")
M. Znojil,
Comment on ``Supersymmetry and Singular Potentials" by Das and Pernice [Nucl. Phys. B 561 (1999) 357]
( hep-th/0209262v2), Nucl. Phys. B 662, 554-562 (2003)
M. Znojil,
``PT-symmetry and supersymmetry",
in ``GROUP 24: Physical and Mathematical Aspects of Symmetries" (IOP Publishing, Bristol, 2003). pp. 629 - 632
(proceedings
of the XXIV International Colloquium on Group Theoretical Methods in Physics, Paris, July 15-20, 2002, Institute of Physics Conference Series Nr. 173, Sect. 7, Ed. Jean-Pierre Gazeau, Richard Kerner, Jean-Pierre Antoine, Stéphane Métens and Jean-Yves Thibon, preprint hep-th/0209062)

M. Znojil,
Expansions in parameter 1/L in PT symmetric quantum theory
(in Czech, proceedings available on CD)
The XIVth conf. of Czech and Slovak physicists (West Bohemia University, Plzen', 9. - 12. Sept. 2002)
O. Mustafa and M. Znojil,
PT symmetric pseudo-perturbation recipe: an imaginary cubic oscillator with spikes
(math-ph/0206042), J. Phys. A: Math. Gen. 35 (2002) 8929 - 8942.
G. Levai and M. Znojil,
The interplay of supersymmetry and ${\cal PT}$ symmetry in quantum mechanics: a case study for the Scarf II potential
(quant-ph/0206013), J. Phys. A: Math. Gen. 35 (2002) 8793 - 8804.
M. Znojil,
A generalization of the concept of PT symmetry
(math-ph/0106021), in ``Quantum Theory and Symmetries", ed. E. Kapuscik and A. Horzela, Word Sci., Singapore,2002, pp. 626-631.
M. Znojil,
Should PT symmetric quantum mechanics be interpreted as nonlinear?
(quant-ph/0103054v4), J. Nonlin. Math. Phys. 9, suppl. 2 (2002), 122-133
(= special issue on Lie Symmetry Analysis and Applications, in honour of the 60th birthday of Peter Leach).
M. Znojil, F. Gemperle and O. Mustafa,
Asymptotic solvability of an imaginary cubic oscillator with spikes
(hep-th/0205181), J. Phys. A: Math. Gen. 35 (2002) 5781 - 5793.
M. Znojil,
Solvable PT-symmetric Hamiltonians
(quant-ph/0008125), Physics of Atomic Nuclei 65 (2002) 1149 - 1151
with parallel Russian version
Yad. Fiz. 65 (2002), 1182-1184.
M. Znojil,
Non-Hermitian SUSY and singular, PT-symmetrized oscillators
(hep-th/0201056), J. Phys. A: Math. Gen. 35 (2002) 2341 - 2352.
M. Znojil and G. Lévai,
Spontaneous breakdown of PT symmetry in the solvable square well model
(hep-th/0111213) Mod. Phys. Letters A 16 (2001) 2273 - 2280.
M. Znojil, G. Le'vai, P. Roy and R. Roychoudhury,
Anomalous doublets of states in a PT symmetric quantum model,
(quant-ph/0104059) Phys. Lett. A 290 (2001) 249 - 254.
G. Lévai and M. Znojil,
Conditions for complex spectra in a class of PT symmetric potentials
(quant-ph/0110064) Mod. Phys. Letters A 16 (2001) 1973 - 1981
B. Bagchi, C. Quesne and M. Znojil,
Generalized continuity equation and modified normalization in PT-symmetric quantum mechanics
(quant-ph/0108096) Mod. Phys. Letters A 16 (2001) 2047 - 2057
M. Znojil,
Poeschl-Teller paradoxes
(math-ph/0102034), J. Phys. A: Math. Gen. 34 (2001) 9585 - 9592.
M. Znojil,
PT symmetric square well
(quant-ph/0101131), Phys. Lett. A 285 (2001) 7 - 10.
M. Znojil,
PT symmetrized SUSY quantum mechanics
(hep-ph/0101038), Czechoslovak J. Phys. 51 (2001) 420 - 428.
M. Znojil and M. Tater,
Complex Calogero model with real energies
(quant-ph/0010087), J. Phys. A: Math. Gen. 34 (2001) 1793-1803.
M. Znojil,
Eight exactly solvable complex potentials in Bender - Boettcher quantum mechanics
Rendiconti del Circ. Mat. di Palermo, Serie II, Suppl. 66 (2001) 213 - 218.
G. Levai and M. Znojil,
Systematic search for PT symmetric potentials with real energy spectra
J. Phys. A: Math. Gen. 33 (2000) 7165 - 7180.
M. Znojil,
Spiked and PT-symmetrized decadic potentials supporting elementary N-plets of bound states
(quant-ph/0002083), J. Phys. A: Math. Gen. 33 (2000) 6825 - 6833.
M. Znojil and G. Levai,
The Coulomb - harmonic-oscillator correspondence in PT symmetric quantum mechanics
(quant-ph/0003081), Phys. Lett. A 271 (2000) 327 - 333.
M. Znojil,
PT-symmetrically regularized Eckart, Poeschl-Teller and Hulthen potentials
(quant-ph/9912027v2), J. Phys. A: Math. Gen. 33 (2000) 4561 - 4572.
M. Znojil, F. Cannata, B. Bagchi and R. Roychoudhury,
Supersymmetry without hermiticity within PT symmetric quantum mechanics
(hep-th/0003277), Phys. Lett. B 483 (2000) 284 - 289.
M. Znojil,
Quasi-exactly solvable quartic potentials with centrifugal and Coulombic terms
(math-ph/0002036v2), J. Phys. A 33 (2000) 4203 - 11.
M. Znojil,
Shape invariant potentials with PT symmetry
(quant-ph/9911116v1), J. Phys. A: Math. Gen. 33 (2000) L61 - L62.
M. Znojil,
Exact solution for Morse oscillator in PT-symmetric quantum mechanics
Phys. Lett. A 264 (1999) 108 - 11.
M. Znojil,
Non-Hermitian matrix description of the PT-symmetric anharmonic oscillators
J. Phys. A: Math. Gen. 32 (1999) 7419 - 28.
M. Znojil,
PT symmetric harmonic oscillators
Phys. Lett. A 259 (1999) 220 - 3.
M. Znojil,
Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable.
J. Phys. A: Math. Gen. 32 (1999) 4563 - 70.
M. F. Fernández, R. Guardiola, J. Ros and M. Znojil,
A family of complex potentials with real spectrum.
J. Phys. A 32 (1999) 3105 - 16.
M. F. Fernández, R. Guardiola, J. Ros and M. Znojil,
Strong-coupling expansions for the PT-symmetric oscillators V(x) = a x + b (ix)^2 + c (ix)^3.
J. Phys. A 31 (1998) 10105 - 12.

ON PT-SYMMETRIC HAMILTONIANS

co-authors:

remark:

contents (click to)

A. PRINCIPLES

a. Origins: real energies from perturbation series

b. Concepts: numerical experiments and explanations

c. Harmonic oscillator: a solvable example

B. TECHNIQUES

d. SUSY: early applications

e. Changes of variables: a proliferation of non-numerical models

f. Square wells: insight and limitations

C. APPLICATIONS

g. Quasi-exact solvability: new horizons

h. Periodic boundary conditions: beyond single particle/dimension

ch. Klein-Gordon equation: advanced applications in physics

a. Origins: real energies from perturbation series

STRONG-COUPLING SERIES

1/L EXPANSIONS

b. Concepts: numerical experiments and explanations

THE WEDGE-DEPENDENCE OF SPECTRA:

HILL-DETERMINANT METHOD - A PROOF:

PARTITIONING AND EFFECTIVE HAMILTONIANS:

APPLICATIONS OUTSIDE QUANTUM THEORY:

c. Harmonic oscillator: a solvable example

COMPLEX SHIFT OF THE COORDINATE:

SUPERSYMMETRIC INTERPRETATION:

d. Supersymmetry: early applications

SHAPE INVARIANCE:

THE VIOLATION OF HERMITICITY:

DIFFERENTIAL CHARGE OPERATORS:

e. Changes of variables: a proliferation of non-numerical models

LAGUERRE POLYNOMIALS:

JACOBI POLYNOMIALS:

ALL HYPERGEOMETRIC POLYNOMIALS:

ANOMALIES:

f. Square wells: insight and limitations

SINGLE MATCHING POINT:

MORE MATCHING POINTS:

g. Quasi-exact solvability: new horizons

ADDING A WEAK, COULOMBIC SPIKE:

CENTRIFUGAL SPIKE IN NEW ROLE:

h. Periodic boundary conditions: beyond single particle/dimension

SOLVABLE MODELS WITH MORE PARTICLES:

PERIODIC BOUNDARY CONDITIONS:

ANGULAR SCHROEDINGER EQUATIONS:

ch. Klein-Gordon equation: advanced applications in physics

FESHBACH - VILLARS RELATIVISTIC HAMILTONIAN:

ENERGY-DEPENDENT INTERACTIONS:

SUPERSYMMETRIC RELATIVISTIC MODELS:

THE END OF THE STORY.

APPENDIX.

SAME MATERIAL, TIME-ORDERED