Non-Stationary Non-Hermitian "Wrong-Sign" Quantum Oscillators and Their Meaningful Physical Interpretation.

In the framework of quantum mechanics using quasi-Hermitian operators the standard unitary evolution of a non-stationary but still closed quantum system is only properly described in the non-Hermitian interaction picture (NIP). In this formulation of the theory both the states and the observables vary with time. A few aspects of implementation of this picture are illustrated via the "wrong-sign" quartic oscillators.

Confluences of exceptional points and a systematic classification of quantum catastrophes.

In the problem of classification of the parameter-controlled quantum phase transitions, attention is turned from the conventional manipulations with the energy-level mergers at exceptional points to the control of mergers of the exceptional points themselves. What is obtained is an exhaustive classification which characterizes every phase transition by the algebraic and geometric multiplicity of the underlying confluent exceptional point. Typical qualitative characteristics of non-equivalent phase transitions are illustrated via a few elementary toy models.

Quantum phase transitions mediated by clustered non-Hermitian degeneracies.

The phenomenon of degeneracy of an N-plet of bound states is studied in the framework of the quasi-Hermitian (a.k.a.

Theory of Response to Perturbations in Non-Hermitian Systems Using Five-Hilbert-Space Reformulation of Unitary Quantum Mechanics.

Non-Hermitian quantum-Hamiltonian-candidate combination H λ of a non-Hermitian unperturbed operator H = H 0 with an arbitrary "small" non-Hermitian perturbation λ W is given a mathematically consistent unitary-evolution interpretation. The formalism generalizes the conventional constructive Rayleigh-Schrödinger perturbation expansion technique. It is sufficiently general to take into account the well known formal ambiguity of reconstruction of the correct physical Hilbert space of states.

Unitary unfoldings of a Bose-Hubbard exceptional point with and without particle number conservation.

The conventional non-Hermitian but -symmetric three-parametric Bose-Hubbard Hamiltonian (, , ) represents a quantum system of bosons, unitary only for parameters , and in a domain . Its boundary contains an exceptional point of order (EPK;  =  + 1) at  = 0 and  = , but even at the smallest non-vanishing parameter  ≠ ~0 the spectrum of (, , ) ceases to be real, i.e.

Quantum phase transitions in nonhermitian harmonic oscillator.

The Stone theorem requires that in a physical Hilbert space [Formula: see text] the time-evolution of a stable quantum system is unitary if and only if the corresponding Hamiltonian H is self-adjoint. Sometimes, a simpler picture of the evolution may be constructed in a manifestly unphysical Hilbert space [Formula: see text] in which H is nonhermitian but [Formula: see text]-symmetric. In applications, unfortunately, one only rarely succeeds in circumventing the key technical obstacle which lies in the necessary reconstruction of the physical Hilbert space [Formula: see text].

The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics.

We propose a unique way to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimizing a 'Hilbert-Schmidt distance' to the original inner product among the entire class of admissible inner products. We prove that either the minimizer exists and is unique or it does not exist at all.