- Correa, F., Fring, A. and Taira, T. (2021).
**Complex BPS Skyrmions with real energy.***Nuclear Physics B*,*971*, pp. 115516–115516. doi:10.1016/j.nuclphysb.2021.115516. - Fring, A. and Tenney, R. (2021).
**Exactly solvable time-dependent non-Hermitian quantum systems from point transformations.***Physics Letters A*,*410*, pp. 127548–127548. doi:10.1016/j.physleta.2021.127548. - Fring, A. and Whittington, S. (2021).
**Lorentzian Toda field theories.***Reviews in Mathematical Physics*,*33*(6). doi:10.1142/s0129055x21500173. - Fring, A. and Tenney, R. (2021).
**Perturbative approach for strong and weakly coupled time-dependent for non-Hermitian quantum systems.***Physica Scripta*,*96*(4), pp. 45211–45211. doi:10.1088/1402-4896/abe259. - Fring, A. and Taira, T. (2020).
**Complex BPS solitons with real energies from duality.***Journal of Physics A: Mathematical and Theoretical*,*53*, pp. 455701–455701. doi:10.1088/1751-8121/abb92a. - Fring, A. and Taira, T. (2020).
**'t Hooft-Polyakov monopoles in non-Hermitian quantum field theory.***Physics Letters B*,*807*, pp. 135583–135583. doi:10.1016/j.physletb.2020.135583. - Fring, A. and Whittington, S. (2020).
**n-Extended Lorentzian Kac–Moody algebras.***Letters in Mathematical Physics*,*110*(7), pp. 1689–1710. doi:10.1007/s11005-020-01272-2. - Fring, A. and Tenney, R. (2020).
**Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators.***Physics Letters A*,*384*(21), pp. 126530–126530. doi:10.1016/j.physleta.2020.126530. - Cen, J., Correa, F. and Fring, A. (2020).
**Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau–Lifschitz equation.***Journal of Physics A: Mathematical and Theoretical*,*53*(19), pp. 195201–195201. doi:10.1088/1751-8121/ab81d9. - Fring, A. and Frith, T. (2020).
**Time-dependent metric for the two-dimensional, non-Hermitian coupled oscillator.***Modern Physics Letters A*,*35*(08), pp. 2050041–2050041. doi:10.1142/s0217732320500418. - Fring, A. and Taira, T. (2020).
**Pseudo-Hermitian approach to Goldstone’s theorem in non-Abelian non-Hermitian quantum field theories.***Physical Review D*,*101*(4). doi:10.1103/physrevd.101.045014. - Fring, A. and Tenney, R. (2020).
**Time-independent approximations for time-dependent optical potentials.***The European Physical Journal Plus*,*135*(2). doi:10.1140/epjp/s13360-020-00143-y. - Cen, J. and Fring, A. (2020).
**Multicomplex solitons.***Journal of Nonlinear Mathematical Physics*,*27*(1), pp. 17–35. doi:10.1080/14029251.2020.1683963. - Fring, A. and Taira, T. (2020).
**Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories.***Nuclear Physics B*,*950*, pp. 1–1. doi:10.1016/j.nuclphysb.2019.114834. - Cen, J. and Fring, A. (2019).
**Asymptotic and scattering behaviour for degenerate multi-solitons in the Hirota equation.***Physica D: Nonlinear Phenomena*,*397*, pp. 17–24. doi:10.1016/j.physd.2019.05.005. - Cen, J., Correa, F. and Fring, A. (2019).
**Integrable nonlocal Hirota equations.***Journal of Mathematical Physics*,*60*(8), pp. 81508–81508. doi:10.1063/1.5013154. - Fring, A. and Frith, T. (2019).
**Eternal life of entropy in non-Hermitian quantum systems.***Physical Review A*,*100*(1). doi:10.1103/physreva.100.010102. - Cen, J., Fring, A. and Frith, T. (2019).
**Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems.***Journal of Physics A: Mathematical and Theoretical*,*52*(11), pp. 115302–115302. doi:10.1088/1751-8121/ab0335. - Bagchi, B. and Fring, A. (2019).
**Quantum, noncommutative and MOND corrections to the entropic law of gravitation.***International Journal of Modern Physics B*,*33*(05), pp. 1950018–1950018. doi:10.1142/s0217979219500188. - Fring, A. and Frith, T. (2018).
**Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians.***Physics Letters A*. doi:10.1016/j.physleta.2018.10.043. - Fring, A. and Frith, T. (2018).
**Metric versus observable operator representation, higher spin models.***The European Physical Journal Plus*,*133*(2). doi:10.1140/epjp/i2018-11892-4. - (2018).
**Coherent States and Their Applications.**. doi:10.1007/978-3-319-76732-1. - Fring, A. and Moussa, M.H.Y. (2016).
**Non-Hermitian Swanson model with a time-dependent metric.***Physical Review A*,*94*(4). doi:10.1103/physreva.94.042128. - Khantoul, B. and Fring, A. (2015).
**Time-dependent massless Dirac fermions in graphene.***Physics Letters A*,*379*(42), pp. 2704–2706. doi:10.1016/j.physleta.2015.08.011. - Bagarello, F. and Fring, A. (2015).
**Generalized Bogoliubov transformations versus D-pseudo-bosons.***Journal of Mathematical Physics*,*56*(10), pp. 103508–103508. doi:10.1063/1.4933242. - Fring, A. (2015).
**A new non-Hermitian E2-quasi-exactly solvable model.***Physics Letters A*,*379*(10-11), pp. 873–876. doi:10.1016/j.physleta.2015.01.008. - Dey, S., Fring, A. and Mathanaranjan, T. (2014).
**Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type.***International Journal of Theoretical Physics*,*54*(11), pp. 4027–4033. doi:10.1007/s10773-014-2447-4. - Fring, A. and Dey, S. (2013).
**The Two-dimensional Harmonic Oscillator on a Noncommutative Space with Minimal Uncertainties.***Acta Polytechnica: journal of advanced engineering*,*53*, pp. 268–276. - Dey, S., Fring, A. and Gouba, L. (2012).
**PT-symmetric non-commutative spaces with minimal volume uncertainty relations.***Journal of Physics A: Mathematical and Theoretical*,*45*(38). doi:10.1088/1751-8113/45/38/385302. - Fring, A. and Smith, M. (2011).
**PT Invariant Complex E (8) Root Spaces.***INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS*,*50*(4), pp. 974–981. doi:10.1007/s10773-010-0542-8. - Fring, A., Gouba, L. and Bagchi, B. (2010).
**Minimal areas from q-deformed oscillator algebras.***J. Phys. A: Math. Theor.43:425202, 2010*. doi:10.1088/1751-8113/43/42/425202. - Assis, P.E.G. and Fring, A. (2010).
**Compactons versus solitons.***Pramana - Journal of Physics*,*74*(6), pp. 857–865. doi:10.1007/s12043-010-0078-8. - Bagchi, B. and Fring, A. (2009).
**Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems.***PHYS LETT A*,*373*(47), pp. 4307–4310. doi:10.1016/j.physleta.2009.09.054. - Assis, P.E.G. and Fring, A. (2009).
**From real fields to complex Calogero particles.***J PHYS A-MATH THEOR*,*42*(42). doi:10.1088/1751-8113/42/42/425206. - Fring, A. (2009).
**Particles versus fields in PT-symmetrically deformed integrable systems.***PRAMANA-JOURNAL OF PHYSICS*,*73*(2), pp. 363–373. doi:10.1007/s12043-009-0128-2. - Assis, P.E.G. and Fring, A. (2009).
**Integrable models from PT -symmetric deformations.***J PHYS A-MATH THEOR*,*42*(10). doi:10.1088/1751-8113/42/10/105206. - Bagchi, B. and Fring, A. (2009).
**Comment on "Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty".***SYMMETRY INTEGR GEOM*,*5*. doi:10.3842/SIGMA.2009.089. - Bagchi, B. and Fring, A. (2008).
**PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation.***J PHYS A-MATH THEOR*,*41*(39). doi:10.1088/1751-8113/41/39/392004. - Assis, P.E.G. and Fring, A. (2008).
**Metrics and isospectral partners for the most generic cubic PT -symmetric non-Hermitian Hamiltonian.***Journal of Physics A: Mathematical and Theoretical*,*41*(24). doi:10.1088/1751-8113/41/24/244001. - Assis, P.E.G. and Fring, A. (2008).
**The quantum brachistochrone problem for non-Hermitian Hamiltonians.***JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*,*41*(24). doi:10.1088/1751-8113/41/24/244002. - Fring, A., Jones, H. and Znojil, M. (2008).
**6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics: Preface.***Journal of Physics A: Mathematical and Theoretical*,*41*(24). doi:10.1088/1751-8121/41/24/240301. - Fring, A. and Znojil, M. (2008).
**PT -symmetric deformations of Calogero models.***Journal of Physics A: Mathematical and Theoretical*,*41*(19). doi:10.1088/1751-8113/41/19/194010. - Castro Alvaredo, O., Fring, A. and Göhmann, F. (2008).
**On the absence of simultaneous reflection and transmission in integrable impurity systems.***Submited to Phys. Lett.*. - Fring, A. (2007).
**PT-symmetry and Integrability.***Acta Polytechnica 47 (2007) 44-49*. - Faria, C.F.M. and Fring, A. (2007).
**Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation.***LASER PHYS*,*17*(4), pp. 424–437. doi:10.1134/S1054660X07040196. - Fring, A. (2007).
**PT -symmetric deformations of the Korteweg-de Vries equation.***Journal of Physics A: Mathematical and Theoretical*,*40*(15), pp. 4215–4224. - Fring, A. and Manojlovic, N. (2006).
**G(2)-Calogero-Moser Lax operators from reduction.***J NONLINEAR MATH PHY*,*13*(4), pp. 467–478. doi:10.2991/jnmp.2006.13.4.1. - Fring, A. (2006).
**A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models.***Modern Physics Letters A*,*21*(8), pp. 691–699. doi:10.1142/S0217732306019682. - Fring, A. and Korff, C. (2006).
**Non-crystallographic reduction of generalized Calogero-Moser models.***J PHYS A-MATH GEN*,*39*(5), pp. 1115–1131. doi:10.1088/0305-4470/39/5/007. - Figueira De Morisson Faria, C. and Fring, A. (2006).
**Time evolution of non-Hermitian Hamiltonian systems.***Journal of Physics A: Mathematical and General*,*39*(29), pp. 9269–9289. doi:10.1088/0305-4470/39/29/018. - De Morisson Faria, C.F. and Fring, A. (2006).
**Isospectral hamiltonians from moyal products.***Czechoslovak Journal of Physics*,*56*(9), pp. 899–908. doi:10.1007/s10582-006-0386-x. - Fring, A. and Korff, C. (2005).
**Affine Toda field theories related to Coxeter groups of noncrystallographic type.***NUCL PHYS B*,*729*(3), pp. 361–386. doi:10.1016/j.nuclphysb.2005.08.044. - Castro-Alvaredo, O. and Fring, A. (2005).
**Chaos in the thermodynamic Bethe ansatz.***PHYS LETT A*,*334*(2-3), pp. 173–179. doi:10.1016/j.physleta.2004.11.009. - Fring, A. (2005).
**Supersymmetric integrable scattering theories with unstable particles.***J HIGH ENERGY PHYS*, (1). doi:10.1088/1126-6708/2005/01/030. - Castro Alvaredo, O. and Fring, A. (2005).
**Integrable models with unstable particles.***Progress in Mathematics*,*237*, p. 59. - Fring, A. and Korff, C. (2004).
**Exactly solvable potentials of Calogero type for q-deformed Coxeter groups.***J PHYS A-MATH GEN*,*37*(45), pp. 10931–10949. doi:10.1088/0305-4470/37/45/012. - Castro-Alvaredo, A. and Fring, A. (2004).
**On vacuum energies and renomalizability in integrable quantum field theories.***NUCL PHYS B*,*687*(3), pp. 303–322. doi:10.1016/j.nuclphysb.2004.04.005. - Castro-Alvaredo, O.A. and Fring, A. (2004).
**Applications of quantum integrable systems.***International Journal of Modern Physics A*,*19*(SUPPL. 2), pp. 92–116. doi:10.1142/S0217751X04020336. - Castro-Alvaredo, O.A., Dreißig, J. and Fring, A. (2004).
**Integrable scattering theories with unstable particles.***European Physical Journal C*,*35*(3), pp. 393–411. - Castro Alvaredo, O. and Fring, A. (2004).
**Universal boundary reflection amplitudes.***Nucl. Phys.*,*B682*, p. 551. doi:10.1016/j.nuclphysb.2004.01.009. - Castro-Alvaredo, O.A. and Fring, A. (2003).
**Breathers in the elliptic sine-Gordon model.***Journal of Physics A: Mathematical and General*,*36*(40), pp. 10233–10249. doi:10.1088/0305-4470/36/40/008. - Castro-Alvaredo, O.A. and Fring, A. (2003).
**Rational sequences for the conductance in quantum wires from affine Toda field theories.***Journal of Physics A: Mathematical and General*,*36*(26). doi:10.1088/0305-4470/36/26/101. - Castro-Alvaredo, O.A., Figueira de Morisson Faria, C. and Fring, A. (2003).
**Relativistic treatment of harmonics from impurity systems in quantum wires.***Physical Review B - Condensed Matter and Materials Physics*,*67*(12). doi:10.1103/PhysRevB.67.125405. - Castro-Alvaredo, O. and Fring, A. (2003).
**From integrability to conductance, impurity systems.***Nuclear Physics B*,*649*(3), pp. 449–490. doi:10.1016/S0550-3213(02)01029-5. - Castro Alvaredo, O. and Fring, A. (2003).
**Conductance from Non-perturbative Methods II.***JHEP ,PRHEP-unesp2002/015; cond-mat/0210592*. - Castro Alvaredo, O. and Fring, A. (2003).
**Conductance from Non-perturbative Methods I.***JHEP ,PRHEP-unesp2002/010; cond-mat/0210599*. - Fring, A. (2002).
**Mutually local fields from form factors.***International Journal of Modern Physics B*,*16*(14-15), pp. 1915–1924. doi:10.1142/S0217979202011639. - Castro Alvaredo, O. and Fring, A. (2002).
**Finite temperature correlation functions from form factors.***Nucl. Phys.*,*B636*, p. 611. - Castro Alvaredo, O. and Fring, A. (2002).
**Unstable particles versus resonances in impurity systems, conductance in quantum wires.***Journal of Physics: Condensed Matter*,*14*. doi:10.1088/0953-8984/14/47/101. - Castro Alvaredo, O. and Fring, A. (2002).
**Scaling functions from q-deformed Virasoro characters.***J. Phys.*,*A35*, p. 609. doi:10.1088/0305-4470/35/3/310. - Fring, A. (2001).
**Thermodynamic Bethe ansatz and form factors for the homogeneous**

sine-Gordon models.*Nato Science Series*,*35*, pp. 139–153. doi:10.1007/978-94-010-0670-5_9. - Castro-Alvaredo, O.A. and Fring, A. (2001).
**Form factors from free fermionic Fock fields, the Federbush model.***Nuclear Physics B*,*618*(3), pp. 437–464. - Castro Alvaredo, O. and Fring, A. (2001).
**Constructing infinite particle spectra.***Phys. Rev.*,*D64*. doi:10.1103/PhysRevD.64.085005. - Castro Alvaredo, O. and Fring, A. (2001).
**Decoupling the SU(N)2-homogeneous sine-Gordon model.***Phys. Rev.*,*D64*. doi:10.1103/PhysRevD.64.085007. - Castro Alvaredo, O. and Fring, A. (2001).
**Renormalization group flow with unstable particles.***Phys. Rev.*,*D63*. - Castro Alvaredo, O. and Fring, A. (2001).
**Identifying the operator content, the Homogeneous sine-Gordon models.***Nucl. Phys.*,*B604*, p. 367. - Fring, A. and Korff, C. (2000).
**Colour valued scattering matrices.***Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics*,*477*(1-3), pp. 380–386. doi:10.1016/S0370-2693(00)00226-4. - Fring, A. and Korff, C. (2000).
**Large and small density approximations to the thermodynamic Bethe ansatz.***Nuclear Physics B*,*579*(3), pp. 617–631. doi:10.1016/S0550-3213(00)00250-9. - Bytsko, A.G. and Fring, A. (2000).
**Factorized combinations of Virasoro characters.***Communications in Mathematical Physics*,*209*(1), pp. 179–205. - Fring, A., Korff, C. and Schulz, B.J. (2000).
**On the universal representation of the scattering matrix of affine toda field theory.***Nuclear Physics B*,*567*(3), pp. 409–453. - Figueira De Morisson Faria, C., Fring, A. and Schrader, R. (2000).
**Existence criteria for stabilization from the scaling behaviour of ionization probabilities.***Journal of Physics B: Atomic, Molecular and Optical Physics*,*33*(8), pp. 1675–1685. - Castro Alvaredo, O., Fring, A. and Korff, C. (2000).
**Form factors of the homogeneous sine-Gordon models.***Phys. Lett.*,*B484*, p. 167. - Castro Alvaredo, O., Fring, A., Korff, C. and Miramontes, J.L. (2000).
**Thermodynamic Bethe ansatz of the homogeneous sine-Gordon models.***Nucl. Phys.*,*B575*, p. 535. - Faria, C.F.D.M., Fring, A. and Schrader, R. (1999).
**Stabilization not for certain and the usefulness of bounds.***Proc. 8-th Int. Conf. on Multiphothon Processes, ed. L.F. DiMauro et.al. (1999) 150*. - Babujian, H., Fring, A., Karowski, M. and Zapletal, A. (1999).
**Exact form factors in integrable quantum field theories: The sine-Gordon model.***Nuclear Physics B*,*538*(3), pp. 535–586. doi:10.1016/S0550-3213(98)00737-8. - Fring, A., Korff, C. and Schulz, B.J. (1999).
**The ultraviolet behaviour of integrable quantum field theories, affine Toda field theory.***Nuclear Physics B*,*549*(3), pp. 579–612. - Figueira De Morisson Faria, C., Fring, A. and Schrader, R. (1999).
**Analytical treatment of stabilization.***Laser Physics*,*9*(1), pp. 379–387. - Bytsko, A.G. and Fring, A. (1999).
**ADE spectra in conformal field theory.***Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics*,*454*(1-2), pp. 59–69. - Bytsko, A.G. and Fring, A. (1998).
**A Note on ADE-Spectra in Conformal Field Theory.***Phys.Lett. B454 (1999) 59-69*. doi:10.1016/S0370-2693(99)00300-7. - Bytsko, A.G. and Fring, A. (1998).
**Thermodynamic Bethe Ansatz with Haldane Statistics.***Nucl.Phys. B532 (1998) 588-608*. doi:10.1016/S0550-3213(98)00531-8. - Bytsko, A.G. and Fring, A. (1998).
**Anyonic interpretation of Virasoro characters and the thermodynamic Bethe ansatz.***Nuclear Physics B*,*521*(3), pp. 573–591. doi:10.1016/S0550-3213(98)00222-3. - Figueira De Morisson Faria, C., Fring, A. and Schrader, R. (1998).
**On the influence of pulse shapes on ionization probability.***Journal of Physics B: Atomic, Molecular and Optical Physics*,*31*(3), pp. 449–464. - Belavin, A.A. and Fring, A. (1997).
**On the fermionic quasi-particle interpretation in minimal models of conformal field theory.***Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics*,*409*(1-4), pp. 199–205. doi:10.1016/S0370-2693(97)00879-4. - Fring, A., Kostrykin, V. and Schrader, R. (1997).
**Ionization probabilities through ultra-intense fields in the extreme limit.***Journal of Physics A: Mathematical and General*,*30*(24), pp. 8599–8610. - Fring, A. (1996).
**Braid relations in affine Toda field theory.***International Journal of Modern Physics A*,*11*(7), pp. 1337–1352. - Fring, A., Kostrykin, V. and Schrader, R. (1996).
**On the absence of bound-state stabilization through short ultra-intense fields.***Journal of Physics B: Atomic, Molecular and Optical Physics*,*29*(23), pp. 5651–5671. - Fring, A. and Koberle, R. (1994).
**Boundary Bound States in Affine Toda Field Theory.***Int.J.Mod.Phys. A10 (1995) 739-752*. - Fring, A. and Köberle, R. (1994).
**Factorized scattering in the presence of reflecting boundaries.***Nuclear Physics B*,*421*(1), pp. 159–172. - Fring, A. and Köberle, R. (1994).
**Affine Toda field theory in the presence of reflecting boundaries.***Nuclear Physics B*,*419*(3), pp. 647–662. - Fring, A., Johnson, P.R., Kneipp, M.A.C. and Olive, D.I. (1994).
**Vertex operators and soliton time delays in affine Toda field theory.***Nuclear Physics B*,*430*(3), pp. 597–614. - Fring, A. (1993).
**Form Factors in Affine Toda Field Theories.**. - Fring, A., Mussardo, G. and Simonetti, P. (1993).
**Form factors for integrable lagrangian field theories, the sinh-Gordon model.***Nuclear Physics, Section B*,*393*(1-2), pp. 413–441. doi:10.1016/0550-3213(93)90252-K. - Fring, A., Mussardo, G. and Simonetti, P. (1993).
**Form factors of the elementary field in the Bullough-Dodd model.***Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics*,*307*(1-2), pp. 83–90. - Fring, A. (1992).
**Couplings in Affine Toda Field Theories.**. - Fring, A. and Olive, D.I. (1992).
**The fusing rule and the scattering matrix of affine Toda theory.***Nuclear Physics B*,*379*(1-2), pp. 429–447. - Fring, A., Liao, H.C. and Olive, D.I. (1991).
**The mass spectrum and coupling in affine Toda theories.***Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics*,*266*(1-2), pp. 82–86.

## Contact details

## Address

City, University of London

Northampton Square

London EC1V 0HB

United Kingdom

## Personal links

## About

### Overview

Professor Andreas Fring studied physics at the Technische Universität München and the University of London. He received a PhD in theoretical physics in 1992 from Imperial College London. He became a member of the Department of Mathematics in 2004, where he was promoted to Reader in 2005 and subsequently to Professor of Mathematical Physics in 2008.

Before joining City University London he held postdoc positions between 1992 and 1994 at the Universidade de São Paulo (São Carlos, Brazil) and the University of Wales (Swansea, UK). From 1994 until 2004 he was Wissenschaftlicher Mitarbeiter in the Department of Physics at the Freie Universität Berlin.**Recent Talks**

- 06/11, Quantum Physics with non-Hermitian operators (Dresden, Germany)

- 09/11, PT Quantum Mechanics Symposium (Heidelberg, Germany)

- 01/12, UK-Japan winter school (Oxford, UK)

### Qualifications

- PhD Theoretical Physics, Imperial College London, United Kingdom, 1992
- MSc Theoretical Physics, Imperial College London, United Kingdom, 1989
- BSc Physics, University of London, United Kingdom, 1988
- Vordiplom Physics, Frauenklinik der Technischen Universität München, Germany, 1986

### Employment

- Professor, City, University of London, 2008 – present
- Reader, City, University of London, 2005 – 2008
- Lecturer, City, University of London, 2004 – 2005
- C1 Research & Teaching Assistant, Free University of Berlin, 1994 – 2004
- Senior Research Assistant, University of Wales, 1993 – 1994
- Research Assistant, University of Sao Paulo, 1992 – 1993
- Research Assistant, Imperial College London, 1992

### Teaching

### Teaching at City University London

- Dynamical Systems [MA3608]

- Mathematical Methods [MA3605]

- Mathematics [AS1051]

- Integrable Systems [MAM611]

- Geometry & Vectors [MA1607] (05-09)

- Programming Excel/VBA [MA1603] (04-05)

- Advanced Certificate in Mathematics and Statistics (04-06)

## Research

### Research interests

His main field of research is mathematical physics with a focus on integrable quantum field theories and quantum mechanics. He co-authored around one hundred articles published in international journals and conference proceedings on topics including the form factor approach to integrable quantum field theories, factorised scattering theory, the thermodynamic Bethe Ansatz, representation theory of Virasoro algebras and Coxeter/Weyl reflection groups, high energy laser physics, pseudo-Hermitian quantum mechanical systems and noncommutative space-time structures.

### Current research

- Integrable quantum field theory

- High intensity laser physics

- Pseudo-Hermitian Hamiltonian systems in quantum physics

### Recent Keynote speeches and lectures

Pseudo-Hermitian Hamiltonians in Quantum Physics XII

July 2013, Koc University, Istanbul (Turkey)

International conference: Pseudo-Hermitian Hamiltonians in Quantum Physics XII pdf slides

Introduction to non-Hermitian Hamiltonian systems with PT symmetry, applications to integrable systems

January 2012, Oxford University (UK)

UK-Japan winter school : pdf slides

### Research students

### Takanobu Taira

**Attendance: **Jan 2019 – present, full-time

**Thesis title: **Non-Hermitian Quantum Field Theories

**Role: **1st Supervisor

### Rebecca Jade Tenney

**Attendance: **Oct 2018 – present, full-time

**Thesis title: **Non-classical and non-Hermitian aspects of electron-electron correlation in strong-field physics

**Role: **1st Supervisor

### Sam Whittington

**Attendance: **Oct 2018 – present, full-time

**Thesis title: **Extensions of integrable quantum field theories based on Lorentzian Kac-Moody algebras

**Role: **1st Supervisor

### Julia Cen

**Attendance: **Oct 2016 – present, full-time

**Thesis title: **Integrable systems with PT-symmetries

**Role: **1st Supervisor

### Thomas Frith

**Attendance: **Oct 2016 – present, full-time

**Thesis title: **Unitary time-evolution in non-Hermitian quantum systems

**Role: **1st Supervisor

### Hamish Forbes

**Attendance: **Oct 2016 – present

**Thesis title: **The Bach equations in spin-coefficient form

**Role: **1st Supervisor

### Andrea Cavaglia

**Attendance: **Oct 2011 – Oct 2015, full-time

**Thesis title: **Nonsemilinear one-dimensional PDEs: analysis of PT deformed models and numerical study of compactons

**Role: **1st Supervisor

**Further information: **This thesis is based on the work done during my PhD studies and is roughly divided in two independent parts. The first part consists of Chapters 1 and 2 and is based on the two papers Cavaglià et al. [2011] and Cavaglià & Fring [2012], concerning the complex PT-symmetric

deformations of the KdV equation and of the inviscid Burgers equation, respectively. The second part of the thesis, comprising Chapters 3 and 4, contains a review and original numerical studies on the properties of certain quasilinear dispersive PDEs in one dimension with compacton solutions.

The subjects treated in the two parts of this work are quite different, however a common theme, emphasised in the title of the thesis, is the occurrence of nonsemilinear PDEs. Such equations are characterised by the fact that the highest derivative enters the equation in

a nonlinear fashion, and arise in the modeling of strongly nonlinear natural phenomena such as the breaking of waves, the formation of shocks and crests or the creation of liquid drops. Typically, nonsemilinear equations are associated to the development of singularities and

non-analytic solutions. Many of the complex deformations considered in the first two chapters

are nonsemilinear as a result of the PT deformation. This is also a crucial feature of the compacton-supporting equations considered in the second part of this work.

This thesis is organized as follows. Chapter 1 contains an introduction to the field of PT-symmetric quantum and classical mechanics, motivating the study of PT-symmetric deformations of classical systems. Then, we review the contents of Cavaglià et al. [2011] where we explore travelling waves in two family of complex models obtained as PT-symmetric deformations of the KdV equation. We also illustrate with many examples the connection between the periodicity of orbits and their invariance under PTsymmetry.

Chapter 2 is based on the paper Cavaglià & Fring [2012] on the PTsymmetric deformation of the inviscid Burgers equation introduced in Bender & Feinberg [2008]. The main original contribution of this chapter is to characterise precisely how the deformation affects the

gradient catastrophe. We also point out some incorrect conclusions of the paper Bender & Feinberg [2008].

Chapter 3 contains a review on the properties of nonsemilinear dispersive PDEs in one space dimension, concentrating on the compacton solutions discovered in Rosenau & Hyman [1993]. After an introduction, we present some original numerical studies on the K(2, 2) and K(4, 4) equations. The emphasis is on illustrating the different type of phenomena exhibited by the solutions to these models. These numerical experiments confirm previous results on the properties of compacton-compacton collisions. Besides, we make some original observations,

showing the development of a singularity in an initially smooth solution.

In Chapter 4 , we consider an integrable compacton equation introduced by Rosenau in Rosenau [1996]. This equation has been previously studied numerically in an unpublished work by Hyman and Rosenau cited in Rosenau [2006]. We present an independent numerical study, confirming the claim of Rosenau [2006] that travelling compacton equations to this equation do not contribute to the initial value problem. Besides, we analyse the local conservation laws of this

equation and show that most of them are violated by any solution having a compact, dynamically evolving support. We confirm numerically that such solutions, which had not been described before, do indeed exist.

Finally, in Chapter 5 we present our conclusions and discuss open problems related to this work.

### Sanjib Dey

**Attendance: **Oct 2011 – Sep 2014, full-time

**Thesis title: **Quantum mechanics and quantum field theory in noncommutative space

**Role: **1st Supervisor

**Further information: **Intuitive arguments involving standard quantum mechanical uncertainty relations suggest that at length scales close to the Planck length, strong gravity effects limit the spatial as well as temporal resolution smaller than fundamental length scale, leading to space-space as well as spacetime uncertainties. Space-time cannot be probed with a resolution beyond this scale i.e. space-time becomes "fuzzy" below this scale, resulting into noncommutative spacetime. Hence it becomes important and interesting to study in detail the structure of such noncommutative spacetimes and their properties, because it not only helps us to improve our understanding of the Planck scale physics but also helps in bridging standard particle physics with physics at Planck scale.

Our main focus in this thesis is to explore different methods of constructing models in these kind of spaces in higher dimensions. In particular, we provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing non-commutative spaces. The representations for the corresponding operators obey algebras whose uncertainty relations lead to minimal length, areas and volumes in phase space, which are in principle natural candidates of many different approaches of quantum gravity. We study some explicit models on these types of non-commutative spaces, in particular, we provide solutions of three dimensional harmonic oscillator as well as its decomposed versions into lower dimensions. Because the solutions are computed in these cases by utilising the standard Rayleigh-Schrodinger perturbation theory, we investigate a method afterwards to construct models in an exact manner. We demonstrate three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically non-commutative model with Poschl-Teller type potential. In many cases the operators are not Hermitian with regard to the standard inner products and that is the reason why we use PT -symmetry and pseudo-Hermiticity property, wherever applicable, to make them self-consistent well designed physical observables. We construct an exact form of the metric operator, which is rare in the literature, and provide Hermitian versions of the non-Hermitian Euclidean Lie algebraic type Hamiltonian systems. We also indicate the region of broken and unbroken PT -symmetry and provide a theoretical treatment of the gain loss behaviour of these types of systems in the unbroken PT -regime, which draws more attention to the experimental physicists in recent days.

Apart from building mathematical models, we focus on the physical implications of noncommutative theories too. We construct Klauder coherent states for the perturbative and nonperturbative noncommutative harmonic oscillator associated with uncertainty relations implying minimal lengths. In both cases, the uncertainty relations for the constructed states are shown to be saturated and thus imply to the squeezed coherent states. They are also shown to satisfy the Ehrenfest theorem dictating the classical like nature of the coherent wavepacket. The quality of those states are further underpinned by the fractional revival structure which compares the quality of the coherent states with that of the classical particle directly. More investigations into the comparison are carried out by a qualitative comparison between the dynamics of the classical particle and that of the coherent states based on numerical techniques. We find the qualitative behaviour to be governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures. We demonstrate these features explicitly for the harmonic oscillator, the Poschl-Teller potential and a Calogero type potential having singularity at the origin, we argue on the fact that the effects are less visible from the mathematical analysis and stress that the method is quite useful for the precession measurement required for the experimental purpose. In the context of complex classical mechanics we also find the claim that "the trajectories of classical particles in complex potential are always closed and periodic when its energy is real, and open when the energy is complex", which is demanded in the literature, is not in general true and we show that particles with complex energies can possess a closed and periodic orbit and particles with real energies can produce open trajectories.

### Monique Smith

**Attendance: **Oct 2009 – Sep 2012, full-time

**Thesis title: **Antilinear deformations of Coxeter groups with application to Hamiltonian systems

**Role: **1st Supervisor

**Further information: **The thesis provides several different systematic methods for constructing complex root spaces that remain invariant under an antilinear transformation. The first method is based on any element of the Weyl group, which is extended to factorizations of the Coxeter element and a reduced Coxeter element thereafter. An antilinear deformation method for the longest element of the Weyl group is provided. The last construction method leads to an alternative construction for q-deformed roots. Completed in 2012.

### Paulo Eduardo Gonçalves de Assis

**Attendance: **Oct 2006 – Sep 2009, full-time

**Thesis title: **Non-Hermitian Hamiltonians in Field Theory

**Role: **1st Supervisor

**Further information: **This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.

## Publications

### Featured publications

- Bender, C.M., Dorey, P.E., Dunning, C., Fring, A., Hook, D.W., Jones, H.F. … Tateo, R. (2019).
**PT Symmetry.**WORLD SCIENTIFIC (EUROPE). ISBN 978-1-78634-595-0. - Dey, S., Fring, A. and Hussin, V. (2018).
**A Squeezed Review on Coherent States and Nonclassicality for Non-Hermitian Systems with Minimal Length.***Springer Proceedings in Physics*(pp. 209–242). Springer International Publishing. - Fring, A. and Frith, T. (2018).
**Solvable two-dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime.***Journal of Physics A: Mathematical and Theoretical*,*51*(26), pp. 265301–265301. doi:10.1088/1751-8121/aac57b. - Cen, J., Correa, F. and Fring, A. (2017).
**Degenerate multi-solitons in the sine-Gordon equation.***Journal of Physics A: Mathematical and Theoretical*,*50*(43), pp. 435201–435201. doi:10.1088/1751-8121/aa8b7e. - Fring, A. and Frith, T. (2017).
**Mending the broken PT-regime via an explicit time-dependent Dyson map.***Physics Letters A*,*381*(29), pp. 2318–2323. doi:10.1016/j.physleta.2017.05.041. - Fring, A., Cen, J. and Correa, F. (2017).
**Time-delay and reality conditions for complex solitons.***Journal of Mathematical Physics*,*58*, pp. 32901–32901. doi:10.1063/1.4978864. - Bagarello, F. and Fring, A. (2017).
**From pseudo-bosons to pseudo-Hermiticity via multiple generalized Bogoliubov transformations.***International Journal of Modern Physics B*,*31*(12), pp. 1750085–1750085. doi:10.1142/S0217979217500850. - Fring, A. and Frith, T. (2017).
**Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians.***Physical Review A*,*95*(1). doi:10.1103/physreva.95.010102. - Dey, S., Fring, A. and Hussin, V. (2017).
**Nonclassicality versus entanglement in a noncommutative space.***International Journal of Modern Physics B*,*31*(01), pp. 1650248–1650248. doi:10.1142/s0217979216502489. - Cen, J. and Fring, A. (2016).
**Complex solitons with real energies.***Journal of Physics A: Mathematical and Theoretical*,*49*(36), pp. 365202–365202. doi:10.1088/1751-8113/49/36/365202. - Correa, F. and Fring, A. (2016).
**Regularized degenerate multi-solitons.***Journal of High Energy Physics*,*2016*(9). doi:10.1007/jhep09(2016)008. - Fring, A. and Moussa, M.H.Y. (2016).
**Unitary quantum evolution for time-dependent quasi-Hermitian systems with nonobservable Hamiltonians.***Physical Review A*,*93*(4). doi:10.1103/physreva.93.042114. - Fring, A. (2016).
**A Unifying E2-Quasi Exactly Solvable Model.**pp. 235–248. doi:10.1007/978-3-319-31356-6_15. - Dey, S., Fring, A. and Gouba, L. (2015).
**Milne quantization for non-Hermitian systems.***Journal of Physics A: Mathematical and Theoretical*,*48*(40), pp. 40–40. doi:10.1088/1751-8113/48/40/40ft01. - Fring, A. (2015).
**E2-quasi-exact solvability for non-Hermitian models.***Journal of Physics A: Mathematical and Theoretical*,*48*(14), pp. 145301–145301. doi:10.1088/1751-8113/48/14/145301. - Dey, S. and Fring, A. (2014).
**Noncommutative quantum mechanics in a time-dependent background.***Physical Review D*,*90*(8). doi:10.1103/physrevd.90.084005. - Dey, S., Fring, A. and Mathanaranjan, T. (2014).
**Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra.***Annals of Physics*,*346*, pp. 28–41. doi:10.1016/j.aop.2014.04.002. - Bender, C.M., Fring, A. and Komijani, J. (2014).
**Nonlinear eigenvalue problems.***Journal of Physics A: Mathematical and Theoretical*,*47*(23), pp. 235204–235204. doi:10.1088/1751-8113/47/23/235204. - Fring, A. and Bagarello, F. (2013).
**A non self-adjoint model on a two dimensional**

noncommutative space with unbound metric.*Physical Review A: Atomic, Molecular and Optical Physics*,*88*, p. 42119. doi:10.1103/PhysRevA.88.042119. - Fring, A. and Dey, S. (2013).
**Bohmian quantum trajectories from coherent states.***Physical Review A: Atomic, Molecular and Optical Physics*,*88*, p. 22116. doi:10.1103/PhysRevA.88.022116. - Fring, A., Dey, S, and Khantoul, B. (2013).
**Hermitian versus non-Hermitian representations for minimal length uncertainty relations.***Journal of Physics A: Mathematical and Theoretical*,*46*, pp. 335304–335304. doi:10.1088/1751-8113/46/33/335304. - Fring, A., Dey, S., Gouba, L. and Castro, P.G. (2013).
**Time-dependent q-deformed coherent states for generalized uncertainty relations.***Physical Review D: Particles, Fields, Gravitation and Cosmology*,*87*. doi:10.1103/PhysRevD.87.084033. - Fring, A. (2013).
**PT-symmetric deformations of integrable models.***Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*,*371*, p. 20120046. doi:10.1098/rsta.2012.0046. - Fring, A. and Smith, M. (2012).
**Non-Hermitian multi-particle systems from complex root spaces.***Journal of Physics A: Mathematical and Theoretical*,*45*(8). doi:10.1088/1751-8113/45/8/085203. - Dey, S. and Fring, A. (2012).
**Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations.***Physical Review D - Particles, Fields, Gravitation and Cosmology*,*86*(6). doi:10.1103/PhysRevD.86.064038. - Cavaglia, A. and Fring, A. (2012).
**PT-symmetrically deformed shock waves.***Journal of Physics A: Mathematical and Theoretical*,*45*(44). doi:10.1088/1751-8113/45/44/444010. - Bender, C., Fring, A., Günther, U. and Jones, H. (2012).
**Quantum physics with non-Hermitian operators.***Journal of Physics A: Mathematical and Theoretical*,*45*(44). doi:10.1088/1751-8113/45/44/440301. - Cavaglia, A., Fring, A. and Bagchi, B. (2011).
**PT-symmetry breaking in complex nonlinear wave equations and their deformations.***J PHYS A-MATH THEOR*,*44*(32). doi:10.1088/1751-8113/44/32/325201. - Fring, A., Gouba, L. and Scholtz, F.G. (2010).
**Strings from position-dependent noncommutativity.***J PHYS A-MATH THEOR*,*43*(34). doi:10.1088/1751-8113/43/34/345401. - Fring, A. and Smith, M. (2010).
**Antilinear deformations of Coxeter groups, an application to Calogero models.***J PHYS A-MATH THEOR*,*43*(32). doi:10.1088/1751-8113/43/32/325201. - Castro-Alvaredo, O.A. and Fring, A. (2009).
**A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field.***J PHYS A-MATH THEOR*,*42*(46). doi:10.1088/1751-8113/42/46/465211. - Assis, P.E.G. and Fring, A. (2009).
**Non-Hermitian Hamiltonians of Lie algebraic type.***J PHYS A-MATH THEOR*,*42*(1). doi:10.1088/1751-8113/42/1/015203.

### Publications by category

### Conference papers and proceedings (7)

- Castro Alvaredo, O. and Fring, A. (2003).
**Integrable models with unstable particles.**July, Faro (Portugal). - Castro Alvaredo, O. and Fring, A. (2002).
**Applications of quantum integrable systems.**September, Moscow (Russia). - Castro Alvaredo, O. and Fring, A. (2002).
**Conductance from Non-perturbative Methods II.**July, São Paulo (Brazil). - Castro Alvaredo, O. and Fring, A. (2002).
**Aspects of locality in the form factor program.**World Scientific, Singapore. - Castro Alvaredo, O. and Fring, A. (2001).
**Mutually local fields from form factors.**December, Edinburgh, UK. - Castro Alvaredo, O. and Fring, A. (2001).
**Mutually local fields from form factors.**October, Tianjin, China. - Castro Alvaredo, O. and Fring, A. (2001).
**Mutually local fields from form factors.**September, Yerevan (Armenia).

### Journal articles (106)

## Professional activities

### Editorial activity (2)

**Advisory board of Journal of Physics A**.**Editorial board ISRN Mathematical Physics**.

### Keynote lecture/speech

**Pseudo-Hermitian Hamiltonians in Quantum Physics XII.**Koc University, Istanbul (Turkey) (2013). International conference: Pseudo-Hermitian Hamiltonians in Quantum Physics XII

pdf slides

### Media appearances (3)

**Video for Faculti Media TV.**Hermitian versus Non-Hermitian representations for minimal length uncertainty relations

Faculti Media video**Interview in The Guardian.**Must all post doc research have impact?

Article in the Guardian**City Review 2011.**City review 2011, academic excellence