On the Role of the Normalization Factors ${\kappa }_n$ and of the Pseudo-Metric $\mathcal{P}\ne {\mathcal{P}}^{†}$ in Crypto-Hermitian Quantum Models

Abstract

Among $\mathcal{P}$-pseudo-Hermitian Hamiltonians $H ={\cal P}^{-1} H^\dagger \cal P}$ with real spectra, the ''weakly pseudo-Hermitian" ones (i.e., those employing non-self-adjoint $\mathcal{P}\ne {\mathcal{P}}^{†}$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever $\mathcal{P}\ne {\mathcal{P}}^{†}$, the current involutive operator of charge $\mathcal{C}$ gets complemented by a nonequivalent alternative involutive quasiparity operator $\mathcal{Q}$. We show how, in this language, the standard quantum mechanics is restored via the two alternative inner products in the physical Hilbert space of states, with $<{\psi }_1|\mathcal{P}\mathcal{Q}|{\psi }_2>=<{\psi }_1|\mathcal{C}\mathcal{P}|{\psi }_2>$.

This is a contribution to the Proc. of the 3-rd Microconference "Analytic and Algebraic Methods III"(June 19, 2007, Prague, Czech Republic), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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• MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator
• Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
• Fragile PT-symmetry in a solvable model
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Cited by in corpus (10)

• Three-Hilbert-Space Formulation of Quantum Mechanics
• Three solvable matrix models of a quantum catastrophe
• Time-dependent quasi-Hermitian Hamiltonians and the unitarity of quantum evolution
• Quasi-Hermitian formulation of quantum mechanics using two conjugate Schrödinger equations
• Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians
• PT-symmetric dynamical confinement: Fermi acceleration, quantum force and Berry phase
• Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models
• Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint
• Some results on the rotated infinitely deep potential and its coherent states
• Identification of the metric for diagonalizable (anti-)pseudo-Hermitian Hamilton operators represented by two-dimensional matrices