Understanding Partial $\mathcal {P}\mathcal {T}$ Symmetry as Weighted Composition Conjugation in Reproducing Kernel Hilbert Space: An application to Non-hermitian Bose-Hubbard Type Hamiltonian in Fock space

A new kind of symmetry behaviour introduced as partial PT -symmetry(henceforth ∂PT ) is investigated in a typical Fock space setting understood as a Reproducing Kernel Hilbert Space (RKHS). The same kind of symmetry is understood for a nonhermitian Bose-Hubbard type Hamiltonian involving two boson operators as well as its eigenstates. The phenomenon of symmetry breaking has also been considered.

[1]  MIHAI PUTINAR.  COMPLEX SYMMETRIC OPERATORS AND APPLICATIONS II , 2007 .

[2]  Stefano Longhi,et al.  PT-symmetric laser absorber , 2010, 1008.5298.

[3]  P. V. Hai,et al.  Complex symmetric differential operators on Fock space , 2018, Journal of Differential Equations.

[4]  D. Brody.  Biorthogonal quantum mechanics , 2013, 1308.2609.

[5]  H. Korsch,et al.  A non-Hermitian symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points , 2008, 0802.3164.

[6]  Mourad E. H. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable , 2005 .

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

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

[9]  V. N. Rodionov.  Non-Hermitian PT$\mathcal {P}\mathcal {T}$-Symmetric Dirac-Pauli Hamiltonians with Real Energy Eigenvalues in the Magnetic Field , 2014, 1409.5412.

[10]  F. Fernández.  Non-Hermitian Hamiltonians and Similarity Transformations , 2015, 1502.02694.

[11]  P. V. Hai,et al.  Complex symmetry of weighted composition operators on the Fock space , 2016 .

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

[13]  G. Zheng,et al.  Time Evolution of Two-States Non-Hermitian Systems , 2021, International Journal of Theoretical Physics.

[14]  Fabio Bagarello,et al.  Non-selfadjoint operators in quantum physics : mathematical aspects , 2015 .

[15]  V. Paulsen,et al.  An Introduction to the Theory of Reproducing Kernel Hilbert Spaces , 2016 .

[16]  H. Harney,et al.  PT symmetry and spontaneous symmetry breaking in a microwave billiard. , 2011, Physical review letters.

[17]  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.

[18]  J. Rubinstein,et al.  Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven with electric currents. , 2007, Physical review letters.

[19]  Stephan Ramon Garcia,et al.  Complex Symmetric Operators and Applications II , 2005 .

[20]  D. Brody,et al.  Thermodynamics of quantum heat bath , 2014, 1406.5780.

[21]  Carl M. Bender,et al.  Coupled Oscillator Systems Having Partial PT Symmetry , 2015, 1503.05725.

[22]  Liang Hao,et al.  Observation of a fast evolution in a parity-time-symmetric system , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[24]  Shachar Klaiman,et al.  Visualization of branch points in PT-symmetric waveguides. , 2008, Physical review letters.