Quasi-Hermitian formulation of quantum mechanics using two conjugate Schrödinger equations

Abstract

In an amended version of non-Hermitian interaction picture we propose to work with the states $\psi (t)$ in a dyadic representation. The control of evolution via two conjugate Schrödiner equations then renders the usual necessity of the construction of the time-dependent inner-product-metric operator $\Theta (t)$ redundant. The primary information about dynamics is assumed carried by a non-Hamiltonian observable (say, $R(t)$). A specific realization of phase transitions is then rendered possible via the Kato's exceptional-point (EP) degeneracy of the eigenvalues of $R(t)$ at the EP time $t=t^{(EP)}$. For illustration a cosmological model is proposed mimicking the unitary-evolution birth of the Universe from an initial quantum Big Bang singularity.

27 pp., 1 figure

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• Quantum singularities in a solvable toy model
• Few-grid-point simulations of Big Bang singularity in quantum cosmology
• Features, paradoxes and amendments of perturbative non-Hermitian quantum mechanics
• Three alternative model-building strategies using quasi-Hermitian time-dependent observables
• Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models