Complementarity versus coordinate transformations: mapping between pseudo-Hermiticity and weak pseudo-Hermiticity

Abstract

\noindent We study the concept of the complementarity, introduced by Bagchi and Quesne in [Phys. Lett. A {\bf 301}, 173 (2002)], between pseudo-Hermiticity and weak pseudo-Hermiticity in a rigorous mathematical viewpoint of coordinate transformations when a system has a position-dependent mass. We first determine, under the modified-momentum, the generating functions identifying the complexified potentials $V_{\pm }(x)$ under both concepts of pseudo-Hermiticity ${\tilde{\eta }}_{+}$ (resp. weak pseudo-Hermiticity ${\tilde{\eta }}_{-}$). We show that the concept of complementarity can be understood and interpreted as a coordinate transformation through their respective generating functions. As consequence, a similarity transformation which implements coordinate transformations is obtained. We show that the similarity transformation is set up as fundamental relationship connecting both ${\tilde{\eta }}_{+}$ and ${\tilde{\eta }}_{-}$. A special factorization ${\eta }_{+}={\eta }_{-}^{†}{\eta }_{-}$ is discussed in the case of a constant mass and some Bäcklund transformations are derived.

References in corpus (25)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• PT-Symmetric Quantum Mechanics
• Pseudo-Hermitian Representation of Quantum Mechanics
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Complex periodic potentials with real band spectra
• Generalized Continuity Equation and Modified Normalization in PT-Symmetric Quantum Mechanics
• Weak pseudo-Hermiticity and antilinear commutant
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• Pseudo-Hermiticity, weak pseudo-Hermiticity and eta-orthogonality condition
• Conceptual Aspects of PT-Symmetry and Pseudo-Hermiticity: A status report
• $\eta$-weak-Pseudo-Hermiticity generators and exact solvability
• Non-Hermitian d-dimensional Hamiltonians with position dependent mass and their $\eta$-pseudo-Hermiticity generators
• Exact solution of the position-dependent effective mass and angular frequency Schrödinger equation: harmonic oscillator model with quantized confinement parameter
• Pseudo-Hermitian versus Hermitian position-dependent-mass Hamiltonians in a perturbative framework
• Non-hermitean hamiltonians with unitary and antiunitary symmetry
• On the Pseudo-Hermiticity of a Class of PT-Symmetric Hamiltonians in One Dimension
• Is Weak Pseudo-Hermiticity Weaker than Pseudo-Hermiticity?
• Non-Hermitian Hamiltonians and similarity transformations
• Pseudo-Hermitian coherent states under the generalized quantum condition with position-dependent mass
• PT-symmetric model with an interplay between kinematical and dynamical non-localities
• Pseudo-Hermitian systems with PT-symmetry: Degeneracy and Krein space
• The effective potential and resummation procedure to multidimensional complex cubic potentials for weak and strong-coupling