Making sense of non-Hermitian Hamiltonians

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space–time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive! Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm ‘ghost’ state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

[1]  H. Jones,et al.  6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics , 2008 .

[2]  N. Hatano,et al.  Some Properties of the Resonant State in Quantum Mechanics and Its Computation , 2007, 0705.1388.

[3]  Tomio Petrosky,et al.  The Liouville Space Extension of Quantum Mechanics , 2007 .

[4]  T. Curtright,et al.  Planar super-Landau models revisited , 2006, hep-th/0612300.

[5]  C. Bender,et al.  Faster than Hermitian quantum mechanics. , 2006, Physical review letters.

[6]  R. Rivers,et al.  Path-integral derivation of the anomaly for the Hermitian equivalent of the complex PT-symmetric quartic Hamiltonian , 2006, hep-th/0610245.

[7]  M. Znojil On a few new quantization recipes using PT -symmetry , 2006 .

[8]  H. B. Geyer,et al.  The Physics of Non-Hermitian Operators , 2006 .

[9]  A. Khare,et al.  Periodic potentials and PT symmetry , 2006 .

[10]  Z. Ahmed PT-symmetry in conventional quantum physics , 2006 .

[11]  Full time nonexponential decay in double-barrier quantum structures , 2006, quant-ph/0605109.

[12]  C. Bender,et al.  Equivalence of a Complex PT -Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly , 2006, hep-th/0605066.

[13]  F. Scholtz,et al.  Moyal products -- a new perspective on quasi-hermitian quantum mechanics , 2006, quant-ph/0602187.

[14]  Toshiaki Tanaka,et al.  Nonlinear pseudo-supersymmetry in the framework of -fold supersymmetry , 2006, quant-ph/0602177.

[15]  K. Hibberd,et al.  A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons , 2006, nlin/0602032.

[16]  C. Bender,et al.  Classical trajectories for complex Hamiltonians , 2006, math-ph/0602040.

[17]  A. Khare,et al.  Complex periodic potentials with a finite number of band gaps , 2006, quant-ph/0602105.

[18]  H. Jones,et al.  Equivalent Hermitian Hamiltonian for the non-Hermitian -x 4 potential , 2006, quant-ph/0601188.

[19]  C. Bender,et al.  Calculation of the hidden symmetry operator for a -symmetric square well , 2006, quant-ph/0601123.

[20]  G. Uhrig,et al.  Hard-core magnons in the S=1/2 Heisenberg model on the square lattice , 2005, cond-mat/0512244.

[21]  H. B. Geyer,et al.  Operator equations and Moyal products–metrics in quasi-Hermitian quantum mechanics , 2005, quant-ph/0512055.

[22]  F. Wegner Flow equations and normal ordering: a survey , 2005, cond-mat/0511660.

[23]  C. Bender,et al.  Semiclassical analysis of a complex quartic Hamiltonian , 2005, quant-ph/0509034.

[24]  S. Weigert An algorithmic test for diagonalizability of finite-dimensional PT-invariant systems , 2005, quant-ph/0506042.

[25]  C. Bender,et al.  symmetric versus Hermitian formulations of quantum mechanics , 2005, hep-th/0511229.

[26]  J. M. Arias,et al.  Continuous unitary transformations in two-level boson systems , 2005, cond-mat/0509721.

[27]  C. Bender,et al.  Dual PT-symmetric quantum field theories , 2005, hep-th/0508105.

[28]  R. Roychoudhury,et al.  LETTER TO THE EDITOR: Pseudo-Hermiticity and some consequences of a generalized quantum condition , 2005, quant-ph/0508073.

[29]  J. Moffat Charge conjugation invariance of the vacuum and the cosmological constant problem [rapid communication] , 2005, hep-th/0507020.

[30]  A. Khare,et al.  PT-Invariant Periodic Potentials with a Finite Number of Band Gaps , 2005, math-ph/0505027.

[31]  B. Samsonov LETTER TO THE EDITOR: SUSY transformations between diagonalizable and non-diagonalizable Hamiltonians , 2005, quant-ph/0503075.

[32]  G. Horowitz,et al.  Holographic description of AdS cosmologies , 2005, hep-th/0503071.

[33]  M. Znojil,et al.  MHD α2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator , 2005, math-ph/0501069.

[34]  C. Bender,et al.  PT-symmetric quantum electrodynamics , 2005, hep-th/0501180.

[35]  S. Dusuel,et al.  Finite-size scaling exponents and entanglement in the two-level BCS model , 2005, cond-mat/0501282.

[36]  C. Bender,et al.  New quasi-exactly solvable sextic polynomial potentials , 2005, quant-ph/0501053.

[37]  C. Bender Introduction to 𝒫𝒯-symmetric quantum theory , 2005, quant-ph/0501052.

[38]  C. Bender,et al.  The C operator in PT-symmetric quantum field theory transforms as a lorentz scalar , 2004, hep-th/0412316.

[39]  S. Dusuel,et al.  Continuous unitary transformations and finite-size scaling exponents in the Lipkin-Meshkov-Glick model , 2004, cond-mat/0412127.

[40]  H. Jones On pseudo-Hermitian Hamiltonians and their Hermitian counterparts , 2004, quant-ph/0411171.

[41]  C. Bender,et al.  Ghost Busting: PT-Symmetric Interpretation of the Lee Model , 2004, hep-th/0411064.

[42]  R. Tateo,et al.  Beyond the WKB approximation in PT-symmetric quantum mechanics , 2004, hep-th/0410013.

[43]  S. Matsumoto,et al.  Nonexponential decay of an unstable quantum system: Small-Q-value s-wave decay , 2004, quant-ph/0408149.

[44]  M. Znojil,et al.  Construction of PT-asymmetric non-Hermitian Hamiltonians with CPT symmetry , 2004, math-ph/0406031.

[45]  K. Shin The potential (iz)m generates real eigenvalues only, under symmetric rapid decay boundary conditions , 2002, hep-ph/0207251.

[46]  Anton Zettl,et al.  Sturm-Liouville theory , 2005 .

[47]  H. B. Geyer,et al.  CPT - conserving Hamiltonians and their nonlinear supersymmetrization using differential charge-operators C , 2004, hep-th/0412211.

[48]  Carl M. Bender,et al.  The Script C operator in Script PScript T-symmetric quantum theories , 2004 .

[49]  M. Znojil Solvable PT-symmetric model with a tunable interspersion of nonmerging levels , 2004, quant-ph/0410196.

[50]  A. Mostafazadeh Pseudo-Hermitian description of PT-symmetric systems defined on a complex contour , 2004, quant-ph/0410012.

[51]  A. Mostafazadeh,et al.  Physical aspects of pseudo-Hermitian and PT-symmetric quantum mechanics , 2004, quant-ph/0408132.

[52]  A. Carlo,et al.  Atomistic theory of transport in organic and inorganic nanostructures , 2004 .

[53]  K. Shin Eigenvalues of PT-symmetric oscillators with polynomial potentials , 2004, math/0407018.

[54]  C. Bender,et al.  Semiclassical calculation of the C operator in PT -symmetric quantum mechanics , 2004, hep-th/0405113.

[55]  S. Dusuel,et al.  The quartic oscillator: a non-perturbative study by continuous unitary transformations , 2004, cond-mat/0405166.

[56]  Gerhard Klimeck,et al.  The discretized Schrödinger equation and simple models for semiconductor quantum wells , 2004 .

[57]  A. Nanayakkara Classical trajectories of 1D complex non-Hermitian Hamiltonian systems , 2004 .

[58]  K. Shin On the shape of spectra for non-self-adjoint periodic Schrödinger operators , 2004, math-ph/0404015.

[59]  C. Bender,et al.  Erratum: Complex Extension of Quantum Mechanics [Phys. Rev. Lett.89, 270401 (2002)] , 2004 .

[60]  C. Bender,et al.  Extension of PT -Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction , 2004, hep-th/0402183.

[61]  A. Khare,et al.  Analytically solvable PT-invariant periodic potentials , 2004, quant-ph/0402106.

[62]  C. Bender,et al.  Scalar quantum field theory with a complex cubic interaction. , 2004, Physical review letters.

[63]  Mark S. Swanson,et al.  Transition elements for a non-Hermitian quadratic Hamiltonian , 2004 .

[64]  P. Roy,et al.  PT symmetry of a conditionally exactly solvable potential , 2004, quant-ph/0401064.

[65]  P. Roy,et al.  New exactly solvable isospectral partners for symmetric potentials , 2003, quant-ph/0312085.

[66]  G. Scolarici,et al.  Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries , 2003, quant-ph/0310106.

[67]  R. Tateo,et al.  A Reality Proof in PT-Symmetric Quantum Mechanics , 2003, hep-th/0309209.

[68]  G. Uhrig,et al.  Spectral properties of the dimerized and frustrated $S=1/2$ chain , 2003, cond-mat/0307678.

[69]  A. Nanayakkara Classical Motion of Complex 2-D Non-Hermitian Hamiltonian Systems , 2004 .

[70]  R. Tateo,et al.  DIFFERENTIAL EQUATIONS AND THE BETHE ANSATZ , 2003, hep-th/0309054.

[71]  Z. Ahmed Pseudo-reality and pseudo-adjointness of Hamiltonians , 2003, quant-ph/0306093.

[72]  G. Uhrig,et al.  The structure of operators in effective particle-conserving models , 2003, cond-mat/0306333.

[73]  S. Weigert Completeness and Orthonormality in PT -symmetric Quantum Systems , 2003, quant-ph/0306040.

[74]  A. Mostafazadeh Exact PT-symmetry is equivalent to Hermiticity , 2003, quant-ph/0304080.

[75]  P. Vogl,et al.  Efficient method for the calculation of ballistic quantum transport , 2003 .

[76]  Z. Ahmed,et al.  Gaussian ensemble of 2 × 2 pseudo-Hermitian random matrices , 2003 .

[77]  Dorje C. Brody,et al.  Must a Hamiltonian be Hermitian , 2003, hep-th/0303005.

[78]  Z. Ahmed An ensemble of non-Hermitian Gaussian-random 2×2 matrices admitting the Wigner surmise , 2003 .

[79]  Z. Ahmed C-, PT- and CPT-invariance of pseudo-Hermitian Hamiltonians , 2003, quant-ph/0302141.

[80]  C. Handy,et al.  Moment problem quantization within a generalized scalet-Wigner (auto-scaling) transform representation , 2003 .

[81]  Z. Ahmed P-, T-, PT-, and CPT-invariance of Hermitian Hamiltonians , 2003, quant-ph/0302084.

[82]  A. Mostafazadeh Erratum: Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians [J. Math. Phys. 43, 6343 (2002); math-ph/0207009] , 2003, math-ph/0301030.

[83]  Z. Ahmed,et al.  Pseudounitary symmetry and the Gaussian pseudounitary ensemble of random matrices. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[84]  A. Carlo TOPICAL REVIEW: Microscopic theory of nanostructured semiconductor devices: beyond the envelope-function approximation , 2003 .

[85]  C. Knetter Perturbative Continuous Unitary Transformations: Spectral Properties of Low Dimensional Spin Systems , 2003 .

[86]  C. Bender,et al.  Calculation of the hidden symmetry operator in -symmetric quantum mechanics , 2002, quant-ph/0211166.

[87]  A. Mostafazadeh Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries , 2002, math-ph/0209018.

[88]  G. Uhrig,et al.  Renormalization by continuous unitary transformations: one-dimensional spinless fermions , 2002, cond-mat/0208446.

[89]  C. Bender,et al.  Complex extension of quantum mechanics. , 2002, Physical review letters.

[90]  M. Berry,et al.  Generalized PT symmetry and real spectra , 2002 .

[91]  A. Mostafazadeh Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians , 2002, math-ph/0207009.

[92]  B. Bagchi,et al.  Pseudo-Hermiticity, weak pseudo-Hermiticity and η-orthogonality condition , 2002, quant-ph/0206055.

[93]  A. Ventura,et al.  PT symmetry breaking and explicit expressions for the pseudo-norm in the Scarf II potential , 2002, quant-ph/0206032.

[94]  B. Bagchi,et al.  PT-SYMMETRIC SQUARE WELL AND THE ASSOCIATED SUSY HIERARCHIES , 2002, quant-ph/0205003.

[95]  L. Solombrino Weak pseudo-Hermiticity and antilinear commutant , 2002, quant-ph/0203101.

[96]  V. Barsegov,et al.  Quantum decoherence, Zeno process, and time symmetry breaking. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[97]  Z. Ahmed Pseudo-Hermiticity of Hamiltonians under gauge-like transformation: real spectrum of non-Hermitian Hamiltonians , 2002 .

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

[99]  B. Bagchi,et al.  PT symmetric nonpolynomial oscillators and hyperbolic potential with two known real eigenvalues in a SUSY framework , 2002, quant-ph/0201063.

[100]  K. Shin,et al.  On the Reality of the Eigenvalues for a Class of -Symmetric Oscillators , 2002, math-ph/0201013.

[101]  R. Bousso,et al.  Conformal vacua and entropy in de Sitter space , 2001, hep-th/0112218.

[102]  G. Uhrig,et al.  Landau's quasiparticle mapping: Fermi liquid approach and Luttinger liquid behavior. , 2001, Physical review letters.

[103]  B. Bagchi,et al.  Complexified PSUSY and SSUSY interpretations of some PT symmetric Hamiltonians possessing two series of real energy eigenvalues , 2001, quant-ph/0106021.

[104]  Extension of a spectral bounding method to the PT-invariant states of the −(iX)N non-Hermitian potential , 2001 .

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

[106]  M. Znojil,et al.  Conditions for complex spectra in a class of PT symmetric potentials , 2001, quant-ph/0110064.

[107]  M. Znojil,et al.  Generalized continuity equation and modified normalization in PT-symmetric quantum mechanics , 2001, quant-ph/0108096.

[108]  C. Bender,et al.  Bound States of Non-Hermitian Quantum Field Theories , 2001, hep-th/0108057.

[109]  A. Mostafazadeh Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian , 2001, math-ph/0107001.

[110]  Gerhard Klimeck,et al.  Electromagnetic coupling and gauge invariance in the empirical tight-binding method , 2001 .

[111]  Quantum transitions and dressed unstable states , 2001 .

[112]  C. Handy Generating converging bounds to the (complex) discrete states of the P2 + iX3 + iαX Hamiltonian , 2001, math-ph/0104036.

[113]  C. Bender,et al.  Comment on a recent paper by Mezincescu , 2001 .

[114]  G. Japaridze Space of state vectors in -symmetric quantum mechanics , 2001 .

[115]  M. Znojil Conservation of pseudo-norm in PT symmetric quantum mechanics , 2001, math-ph/0104012.

[116]  R. Tateo,et al.  Bethe Ansatz equations , and reality properties in PT-symmetric quantum mechanics , 2022 .

[117]  B. Bagchi,et al.  Generating complex potentials with real eigenvalues in supersymmetric quantum mechanics , 2001, quant-ph/0102093.

[118]  Wen-Wei Lin,et al.  Eigenvalue Problems for One-Dimensional Discrete Schrödinger Operators with Symmetric Boundary Conditions , 2001, SIAM J. Matrix Anal. Appl..

[119]  S. Y. Lou,et al.  Revisitation of the localized excitations of the (2+1)-dimensional KdV equation , 2001 .

[120]  C. Bender,et al.  Calculation of the one point Green's function for a g phi 4 quantum field theory , 2001 .

[121]  M. Znojil PT-symmetrized supersymmetric quantum mechanics , 2001, hep-ph/0101038.

[122]  K. Shin On the eigenproblems of PT-symmetric oscillators , 2000, math-ph/0007006.

[123]  Eric Delabaere,et al.  Spectral analysis of the complex cubic oscillator , 2000 .

[124]  P. Vogl,et al.  Model of room-temperature resonant-tunneling current in metal'insulator and insulator'insulator heterostructures , 2000 .

[125]  C. Bender,et al.  Conjecture on the interlacing of zeros in complex Sturm-Liouville problems , 2000, math-ph/0005012.

[126]  A. Garg Tunnel splittings for one-dimensional potential wells revisited , 2000, cond-mat/0003115.

[127]  G. Mezincescu Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant , 2000, quant-ph/0002056.

[128]  C. Bender,et al.  Solution of Schwinger-Dyson equations for PT symmetric quantum field theory , 1999, hep-th/9907045.

[129]  G. Uhrig,et al.  Perturbation theory by flow equations: dimerized and frustrated S = 1/2 chain , 1999, cond-mat/9906243.

[130]  C. Bender,et al.  Complex square well - a new exactly solvable quantum mechanical model , 1999, quant-ph/9906057.

[131]  C. Bender,et al.  A nonunitary version of massless quantum electrodynamics possessing a critical point , 1999 .

[132]  A. Voros Exact resolution method for general 1D polynomial Schrödinger equation , 1999, math-ph/9902016.

[133]  F. Fernández,et al.  A FAMILY OF COMPLEX POTENTIALS WITH REAL SPECTRUM , 1998, quant-ph/9812026.

[134]  C. Bender,et al.  PT-symmetric quantum mechanics , 1998, 2312.17386.

[135]  Orsay,et al.  SUSY Quantum Mechanics with Complex Superpotentials and Real Energy Spectra , 1998, quant-ph/9806019.

[136]  Éric Delabaere,et al.  Eigenvalues of complex Hamiltonians with PT-symmetry. II , 1998 .

[137]  L. Chebotarev Extensions of the Bohr–Sommerfeld formula to double-well potentials , 1998 .

[138]  Hiroki Nakamura,et al.  Siegert pseudostate formulation of scattering theory: Two-channel case , 1998 .

[139]  F. Cannata,et al.  Schrodinger operators with complex potential but real spectrum , 1998, quant-ph/9805085.

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

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

[142]  C. Bender,et al.  Model of supersymmetric quantum field theory with broken parity symmetry , 1997, hep-th/9710076.

[143]  J. Stein,et al.  Flow equations and the strong-coupling expansion for the Hubbard model , 1997 .

[144]  N. Hatano,et al.  Vortex pinning and non-Hermitian quantum mechanics , 1997, cond-mat/9705290.

[145]  C. Bender,et al.  Semiclassical analysis of quasiexact solvability , 1996, hep-th/9609193.

[146]  Boykin,et al.  Generalized eigenproblem method for surface and interface states: The complex bands of GaAs and AlAs. , 1996, Physical review. B, Condensed matter.

[147]  T. Berggren Expectation value of an operator in a resonant state , 1996 .

[148]  P. Lenz,et al.  Flow equations for electron-phonon interactions , 1996, cond-mat/9604087.

[149]  I. Prigogine,et al.  Poincaré resonances and the extension of classical dynamics , 1996 .

[150]  Nelson,et al.  Localization Transitions in Non-Hermitian Quantum Mechanics. , 1996, Physical review letters.

[151]  S. Datta Electronic transport in mesoscopic systems , 1995 .

[152]  C. Bender,et al.  Quasi‐exactly solvable systems and orthogonal polynomials , 1995, hep-th/9511138.

[153]  P. Neu,et al.  Flow equations for the spin-boson problem , 1995 .

[154]  A. Ushveridze Quasi-Exactly Solvable Models in Quantum Mechanics , 1994 .

[155]  V. Buslaev,et al.  Equivalence of unstable anharmonic oscillators and double wells , 1993 .

[156]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

[157]  C. Bender,et al.  Analytic continuation of eigenvalue problems , 1993 .

[158]  T. Hollowood Solitons in affine Toda field theories , 1991, hep-th/9110010.

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

[160]  K. Rothe,et al.  Non-perturbative methods in 2 dimensional quantum field theory , 1991 .

[161]  R. Brockett Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems , 1991 .

[162]  A. Zamolodchikov Two-point correlation function in scaling Lee-Yang model , 1991 .

[163]  F. Haake Quantum signatures of chaos , 1991 .

[164]  Aharonov,et al.  Geometry of quantum evolution. , 1990, Physical review letters.

[165]  Dunne,et al.  Integration of operator differential equations. , 1989, Physical review. D, Particles and fields.

[166]  Dunne,et al.  Exact solutions to operator differential equations. , 1989, Physical review. D, Particles and fields.

[167]  J. Cardy,et al.  S Matrix of the Yang-Lee Edge Singularity in Two-Dimensions , 1989 .

[168]  C. Bender,et al.  A new perturbative approach to nonlinear problems , 1989 .

[169]  J. J. Sakurai,et al.  Modern Quantum Mechanics , 1986 .

[170]  J. Cardy,et al.  Conformal invariance and the Yang-Lee edge singularity in two dimensions. , 1985, Physical review letters.

[171]  N. David Mermin,et al.  Is the Moon There When Nobody Looks? Reality and the Quantum Theory , 1985 .

[172]  F. R. Gantmakher The Theory of Matrices , 1984 .

[173]  P. Vogl,et al.  A Semi-empirical tight-binding theory of the electronic structure of semiconductors†☆ , 1983 .

[174]  T. Berggren Completeness relations, Mittag-Leffler expansions and the perturbation theory of resonant states , 1982 .

[175]  A. Andrianov The large N expansion as a local perturbation theory , 1982 .

[176]  J. Leray Lagrangian analysis and quantum mechanics : a mathematical structure related to asymptotic expansions and the Maslov index , 1981 .

[177]  E. Caliceti,et al.  Perturbation theory of odd anharmonic oscillators , 1980 .

[178]  B. Harms,et al.  New structure in the energy spectrum of reggeon quantum mechanics with quartic couplings , 1980 .

[179]  W. Romo A study of the completeness properties of resonant states , 1980 .

[180]  B. Harms,et al.  Complex energy spectra in reggeon quantum mechanics with quartic interactions , 1980 .

[181]  Michael E. Fisher,et al.  Yang-Lee Edge Singularity and ϕ 3 Field Theory , 1978 .

[182]  Richard C. Brower,et al.  Critical Exponents for the Reggeon Quantum Spin Model , 1978 .

[183]  E. Sudarshan,et al.  Zeno's paradox in quantum theory , 1976 .

[184]  G. G. Calderón An expansion of continuum wave functions in terms of resonant states , 1976 .

[185]  L. Schiff,et al.  Quantum Mechanics, 3rd ed. , 1973 .

[186]  K. Symanzik Small-distance-behaviour analysis and Wilson expansions , 1971 .

[187]  R. More THEORY OF DECAYING STATES. , 1971 .

[188]  K. Symanzik Small-distance behaviour in field theory , 1971 .

[189]  T. Berggren On a probabilistic interpretation of expansion coefficients in the non-relativistic quantum theory of resonant states , 1970 .

[190]  T. D. Lee,et al.  Negative Metric and the Unitarity of the S Matrix , 1969 .

[191]  W. Romo Inner product for resonant states and shell-model applications , 1968 .

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

[193]  P. Goldbart,et al.  Linear differential operators , 1967 .

[194]  N. Hokkyo A Remark on the Norm of the Unstable State : A Role of Adjoint Wave Functions in Non-Self-Adjoint Quantum Systems , 1965 .

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

[196]  A. Wightman,et al.  PCT, spin and statistics, and all that , 1964 .

[197]  G. Barton Introduction to Advanced Field Theory , 1963 .

[198]  Silvan S. Schweber,et al.  An Introduction to Relativistic Quantum Field Theory , 1962 .

[199]  A. Messiah Quantum Mechanics , 1961 .

[200]  R. Newton Analytic Properties of Radial Wave Functions , 1960 .

[201]  N. N. Bogoliubov,et al.  Introduction to the theory of quantized fields , 1960 .

[202]  T. Wu Ground State of a Bose System of Hard Spheres , 1959 .

[203]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .

[204]  R. Landauer,et al.  Spatial variation of currents and fields due to localized scatterers in metallic conduction , 1988, IBM J. Res. Dev..

[205]  L. Spitzer Physics of fully ionized gases , 1956 .

[206]  T. D. Lee,et al.  Some Special Examples in Renormalizable Field Theory , 1954 .

[207]  R. Dicke Coherence in Spontaneous Radiation Processes , 1954 .

[208]  F. Dyson Divergence of perturbation theory in quantum electrodynamics , 1952 .

[209]  K. Bleuler A NEW METHOD FOR THE TREATMENT OF LONGITUDINAL AND SCALAR PHOTONS , 1950 .

[210]  Suraj N. Gupta Theory of longitudinal photons in quantum electrodynamics , 1950 .

[211]  Suraj N. Gupta On the Calculation of Self-Energy of Particles , 1950 .

[212]  J. E. Moyal Quantum mechanics as a statistical theory , 1949, Mathematical Proceedings of the Cambridge Philosophical Society.

[213]  W. Pauli On Dirac's New Method of Field Quantization , 1943 .

[214]  Paul Adrien Maurice Dirac,et al.  Bakerian Lecture - The physical interpretation of quantum mechanics , 1942, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[215]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[216]  A. Siegert On the Derivation of the Dispersion Formula for Nuclear Reactions , 1939 .

[217]  E. Wigner On the quantum correction for thermodynamic equilibrium , 1932 .

[218]  P. Dirac Principles of Quantum Mechanics , 1982 .