Non-stationary non-Hermitian "wrong-sign'' quantum oscillators and their meaningful physical interpretation
Miloslav Znojil
TL;DR
The paper develops the non-Hermitian interaction picture (NIP) for non-stationary closed quantum systems, showing that the instantaneous energy is given by $H(t)=G(t)+Σ(t)$ while $G(t)$ and $Σ(t)$ themselves may have complex spectra. It explains that the spectra of $G(t)$ and $Σ(t)$ are phenomenologically irrelevant and that a meaningful interpretation relies on the amended, time-dependent metric in the physical Hilbert space, with $H† Θ = Θ H$. Through stationary and time-dependent analyses of wrong-sign oscillators, including perturbative and ${\cal PT}$-symmetric constructions and the Jones–Mateo $δ=2$ model, the work demonstrates how real, discrete spectra can be reconciled with non-Hermitian dynamics. The results provide methodological addenda and alternatives for achieving unitary evolution in time-dependent non-Hermitian quantum systems, broadening the toolkit for interpreting non-Hermitian Hamiltonians with real spectra in quantum mechanics.
Abstract
Quantum mechanics of closed, unitary quantum systems can be formulated in non-Hermitian interaction picture (NIP) in which both the states and the observables vary with time. Then, in general, not only the Schrödinger-equation generators $G(t)$ but also the Heisenberg-equation generators $Σ(t)$ are phenomenologically irrelevant, with spectra which are, in general, complex. Only the sum $H(t)=G(t)+Σ(t)$ retains the standard physical meaning of instantaneous energy. For illustration, the ``wrong-sign'' quartic oscillators are recalled and reconsidered.