The physics of exceptional points

A short resume is given about the nature of exceptional points (EPs) followed by discussions about their ubiquitous occurrence in a great variety of physical problems. EPs feature in classical as well as in quantum mechanical problems. They are associated with symmetry breaking for -symmetric Hamiltonians, where a great number of experiments has been performed, in particular in optics, and to an increasing extent in atomic and molecular physics. EPs are involved in quantum phase transition and quantum chaos; they produce dramatic effects in multichannel scattering, specific time dependence and more. In nuclear physics, they are associated with instabilities and continuum problems. Being spectral singularities they also affect approximation schemes.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.

[1]  Hui Cao,et al.  Unidirectional invisibility induced by PT-symmetric periodic structures. , 2011, Physical review letters.

[2]  Tsampikos Kottos,et al.  Experimental study of active LRC circuits with PT symmetries , 2011, 1109.2913.

[3]  I. Rotter,et al.  Projective Hilbert space structures at exceptional points , 2007, 0704.1291.

[4]  F. Scholtz,et al.  Quasi-Hermitian operators in quantum mechanics and the variational principle , 1992 .

[5]  H. Jones,et al.  PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold , 2011, 1104.2838.

[6]  A. Mostafazadeh,et al.  Geometric Phase for Non-Hermitian Hamiltonians and Its Holonomy Interpretation , 2008, 0807.3405.

[7]  F. Busse,et al.  Convection in a rotating cylindrical annulus: thermal Rossby waves , 1986, Journal of Fluid Mechanics.

[8]  H. Lipkin,et al.  Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory , 1965 .

[9]  W. Heiss,et al.  Resonance scattering and singularities of the scattering function , 2010, 1003.3105.

[10]  Jan Wiersig,et al.  Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles , 2011 .

[11]  W. Heiss,et al.  The large N behaviour of the Lipkin model and exceptional points , 2005 .

[12]  Dorje C Brody,et al.  Faster than Hermitian quantum mechanics. , 2007, Physical review letters.

[13]  R. Morandotti,et al.  Observation of PT-symmetry breaking in complex optical potentials. , 2009, Physical review letters.

[14]  Heiss.  Repulsion of resonance states and exceptional points , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  P. Wagner,et al.  Bose-Einstein condensates with attractive 1/r interaction: The case of self-trapping , 2007, 0709.2866.

[16]  E. Graefe,et al.  Signatures of three coalescing eigenfunctions , 2011, 1110.1489.

[17]  M. Berry,et al.  Diffraction by volume gratings with imaginary potentials , 1998 .

[18]  Songky Moon,et al.  Observation of an exceptional point in a chaotic optical microcavity. , 2009, Physical review letters.

[19]  M. Berry.  Pancharatnam, virtuoso of the Poincaré , 1994 .

[20]  Tai Tsun Wu,et al.  Analytic Structure of Energy Levels in a Field-Theory Model , 1968 .

[21]  Ingrid Rotter,et al.  A non-Hermitian Hamilton operator and the physics of open quantum systems , 2009 .

[22]  Heiss,et al.  Transitional regions of finite Fermi systems and quantum chaos. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[23]  Phases of wave functions and level repulsion , 1999, quant-ph/9901023.

[24]  T. Stehmann,et al.  Observation of exceptional points in electronic circuits , 2003 .

[25]  W. Nazarewicz,et al.  Open problems in the theory of nuclear open quantum systems , 2010, 1002.0770.

[26]  S. Friedland,et al.  Association of resonance states with the incomplete spectrum of finite complex-scaled Hamiltonian matrices , 1980 .

[27]  T. Prosen,et al.  PT-symmetric wave Chaos. , 2010, Physical review letters.

[28]  A. Jáuregui,et al.  Degeneracy of resonances in a double barrier potential , 2000 .

[29]  Ragnar Fleischmann,et al.  Exponentially fragile PT symmetry in lattices with localized eigenmodes. , 2009, Physical review letters.

[30]  W. Heiss,et al.  Choice of a metric for the non-Hermitian oscillator , 2006 .

[31]  Daniel W Hook,et al.  Complex correspondence principle. , 2009, Physical review letters.

[32]  H. L. Harney,et al.  The chirality of exceptional points , 2001 .

[33]  The physical interpretation of non-Hermitian Hamiltonians and other observables , 2008 .

[34]  Oleg N. Kirillov,et al.  Exceptional points in a microwave billiard with time-reversal invariance violation. , 2010, Physical review letters.

[35]  Exceptional points in multichannel resonance quantization , 2010 .

[36]  B. McCoy,et al.  Holonomy of the Ising model form factors , 2006, math-ph/0609074.

[37]  H. Lipkin,et al.  VALIDITY OF MANY-BODY APPROXIMATION METHODS FOR A SOLVABLE MODEL. III. DIAGRAM SUMMATIONS , 1965 .

[38]  N. Moiseyev,et al.  On the observability and asymmetry of adiabatic state flips generated by exceptional points , 2011 .

[39]  Harry J. Lipkin,et al.  Validity of many-body approximation methods for a solvable model: (II). Linearization procedures , 1965 .

[40]  B. Bagchi,et al.  -symmetric extensions of the supersymmetric Korteweg–de Vries equation , 2008, 0807.1828.

[41]  W. Nazarewicz,et al.  Shell model in the complex energy plane , 2008, 0810.2728.

[42]  Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians. I. General properties , 2006, math-ph/0610024.

[43]  Oleg N. Kirillov,et al.  Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella? , 2009, 0906.1650.

[44]  H. Weidenmüller,et al.  The effective interaction in nuclei and its perturbation expansion: An algebraic approach , 1972 .

[45]  S. Pancharatnam.  The propagation of light in absorbing biaxial crystals — I. Theoretical , 1955 .

[46]  W. Heiss.  Analytic continuation of a lippmann-schwinger kernel , 1970 .

[47]  H. Harney,et al.  Experimental observation of the topological structure of exceptional points. , 2001, Physical review letters.

[48]  W. Heiss.  Chirality of wavefunctions for three coalescing levels , 2007, 0708.1392.

[49]  Sungsam Kang,et al.  Quasieigenstate coalescence in an atom-cavity quantum composite. , 2010, Physical review letters.

[50]  Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum , 2001, math-ph/0110016.

[51]  Heidelberg,et al.  Encircling an exceptional point. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  Carl M. Bender,et al.  Making sense of non-Hermitian Hamiltonians , 2007, hep-th/0703096.

[53]  Friedrich,et al.  Regularity and irregularity in spectra of the magnetized hydrogen atom. , 1986, Physical review letters.

[54]  S. Heinze,et al.  Coulomb analogy for non-Hermitian degeneracies near quantum phase transitions. , 2007, Physical review letters.

[55]  Stefano Longhi,et al.  Optical realization of relativistic non-Hermitian quantum mechanics. , 2010, Physical review letters.

[56]  Li Ge,et al.  PT-symmetry breaking and laser-absorber modes in optical scattering systems. , 2010, Physical review letters.

[57]  N. Moiseyev,et al.  Resonance coalescence in molecular photodissociation. , 2009, Physical review letters.

[58]  Discovery of exceptional points in the Bose-Einstein condensation of gases with attractive 1 / r interaction , 2007, 0711.4786.

[59]  A. Mostafazadeh.  Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries , 2002, math-ph/0203005.

[60]  W. Heiss,et al.  Time behaviour near to spectral singularities , 2010, 1009.5780.

[61]  W. Heiss,et al.  Self-consistent harmonic oscillator model and tilted rotation , 2002 .

[62]  Heidelberg,et al.  Observation of a chiral state in a microwave cavity. , 2002, Physical review letters.

[63]  Jan Wiersig,et al.  Asymmetric scattering and nonorthogonal mode patterns in optical microspirals , 2008, 0810.1584.

[64]  C. Jones.  MULTIPLE EIGENVALUES AND MODE CLASSIFICATION IN PLANE OISEUILLE FLOW , 1988 .

[65]  A. Mostafazadeh.  Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. , 2009, Physical review letters.

[66]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

[67]  W. Heiss,et al.  Exceptional points of non-Hermitian operators , 2003, quant-ph/0304152.

[68]  N. Moiseyev,et al.  Non-Hermitian Quantum Mechanics: Frontmatter , 2011 .

[69]  Oleg N. Kirillov.  PT-symmetry, indefinite damping and dissipation-induced instabilities , 2011, 1110.0018.

[70]  M. Lepers,et al.  Proposal for a laser control of vibrational cooling in Na(2) using resonance coalescence. , 2011, Physical review letters.

[71]  Li Ge,et al.  Pump-induced exceptional points in lasers. , 2011, Physical review letters.

[72]  H. Harney,et al.  Time reversal and exceptional points , 2004 .

[73]  Z. Musslimani,et al.  PT -symmetric optical lattices , 2010 .

[74]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[75]  M. Strayer,et al.  The Nuclear Many-Body Problem , 2004 .