Theory of response to perturbations in non-Hermitian systems using five-Hilbert-space reformulation of unitary quantum mechanics

Abstract

In conventional Schrödinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h}\to H=\Omega \mathfrak{h}\Omega \ne H^{†}$ induces the change $\mathcal{L}\to \mathcal{K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\ne H^{†}$. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts $\mathcal{K}$ into the third, correct physical Hilbert space $\mathcal{H}$, unitarily equivalent to $\mathcal{L}$. The situation encountered, typically, in $\mathcal{P}\mathcal{T}-$symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.

28 pages

References in corpus (10)

• Making Sense of Non-Hermitian Hamiltonians
• The physics of exceptional points
• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
• Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians
• Unitarity corridors to exceptional points
• Linear representation of energy-dependent Hamiltonians
• Crypto-Hermitian Approach to the Klein-Gordon Equation

Cited by in corpus (10)

• Two-qubit entanglement generation through non-Hermitian Hamiltonians induced by repeated measurements on an ancilla
• Anomalous mechanisms of the loss of observability in non-Hermitian quantum models
• Paths of unitary access to exceptional points
• Non-Hermitian Generalization of Rayleigh-Schrödinger Perturbation Theory
• Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via $\mathcal{P}\mathcal{T}-$symmetric and exactly solvable linear-Bose-Hubbard building blocks
• Quantum singularities in a solvable toy model
• Features, paradoxes and amendments of perturbative non-Hermitian quantum mechanics
• Phase transitions in quasi-Hermitian quantum models at exceptional points of order four
• $\mathcal{P}\mathcal{T}$-Symmetric Generalized Extended Momentum Operator
• Triple exceptional point with unitary paths of unfolding in a three-site fermionic Swanson-like model