Three-Hilbert-Space Formulation of Quantum Mechanics

In paper (Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874) the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as H (auxiliary) and H (standard) ) we spot a weak point of the 2HS formalism which lies in the double role played by H (auxiliary) . As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator H(gen) of the time-evolution of the wave functions may dier from their Hamiltonian H.

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

[2]  Dorje C Brody,et al.  Complex extension of quantum mechanics. , 2002, Physical review letters.

[3]  A. Mostafazadeh.  Time-dependent pseudo-Hermitian Hamiltonians defining a unitary quantum system and uniqueness of the metric operator , 2007, 0706.1872.

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

[5]  Miloslav Znojil,et al.  Strong-coupling expansions for the -symmetric oscillators , 1998 .

[6]  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.

[7]  J. Williams.  Operators similar to their adjoints , 1969 .

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

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

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

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

[12]  M. Znojil.  On the Role of the Normalization Factors $\kappa_n$ and of the Pseudo-Metric ${\cal P} \neq {\cal P}^\dagger$ in Crypto-Hermitian Quantum Models , 2007, 0710.4432.

[13]  Ali Mostafazadeh,et al.  Pseudo-Hermitian Representation of Quantum Mechanics , 2008, 0810.5643.

[14]  A. Mostafazadeh,et al.  Propagation of electromagnetic waves in linear media and pseudo-hermiticity , 2007, physics/0703080.

[15]  A. Fring,et al.  Time evolution of non-Hermitian Hamiltonian systems , 2006, quant-ph/0604014.

[16]  H. Jones.  Scattering from localized non-Hermitian potentials , 2007, 0707.3031.

[17]  A. Mostafazadeh.  Pseudo-Hermitian Quantum Mechanics , 2008 .

[18]  Hilbert space structures on the solution space of Klein-Gordon type evolution equations , 2002, math-ph/0209014.

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

[20]  E. Davies,et al.  Linear Operators and their Spectra , 2007 .

[21]  M. Znojil.  Discrete -symmetric models of scattering , 2008, 0806.2019.

[22]  Scattering in PT-symmetric quantum mechanics , 2006, quant-ph/0606129.

[23]  H. Feshbach,et al.  Elementary Relativistic Wave Mechanics of Spin 0 and Spin 1/2 Particles , 1958 .

[24]  M. Znojil.  Time-dependent version of crypto-Hermitian quantum theory , 2008, 0809.2874.

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

[26]  A. Mostafazadeh,et al.  Quantum mechanics of Proca fields , 2008, 0805.1651.

[27]  Daniel F. Styer,et al.  Nine formulations of quantum mechanics , 2002 .

[28]  A. Messiah.  Quantum Mechanics , 1961 .

[29]  Thermodynamics of Pseudo-Hermitian Systems in Equilibrium , 2007 .

[30]  Which operator generates time evolution in Quantum Mechanics , 2007, 0711.0535.

[31]  Carl M. Bender,et al.  Anharmonic oscillator , 1973 .

[32]  Quantum knots , 2008, 0802.1318.

[33]  M. Znojil.  Identification of observables in quantum toboggans , 2008, 0803.0403.

[34]  R. Tateo,et al.  The ODE/IM correspondence , 2001, hep-th/0703066.

[35]  渋谷 泰隆.  Global theory of a second order linear ordinary differential equation with a polynomial coefficient , 1975 .

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

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