An an exactly solvable quantum model is found to sample the evolution towards the sudden and complete loss of observability. For the purpose we choose an N-level system. While the time runs from 0 to 1, the process (leading to the collapse) is controlled by a toy-model Hamiltonian H and by a unitary-evolution guaranteeing minimally anisotropic (i.e., unique) Hilbert-space metric. The process of the degeneracy of the real N-plet of energy levels is studied without the usual assumption of adiabaticity. The initial Hamiltonian is diagonal, the initial metric is chosen as identity. Once the system reaches the observability horizon, the metric becomes singular (of rank one) while the end-point Hamiltonian acquires the canonical Jordan-block form (i.e., it loses its diagonalizability). An optimal measure of the distance from the final catastrophe is finally found in a universal, exact formula for the spectrum of the metric.
An an exactly solvable quantum model is found to sample the evolution towards the sudden and complete loss of observability. For the purpose we choose an N-level system. While the time runs from 0 to 1, the process (leading to the collapse) is controlled by a toy-model Hamiltonian H and by a unitary-evolution guaranteeing minimally anisotropic (i.e., unique) Hilbert-space metric. The process of the degeneracy of the real N-plet of energy levels is studied without the usual assumption of adiabaticity. The initial Hamiltonian is diagonal, the initial metric is chosen as identity. Once the system reaches the observability horizon, the metric becomes singular (of rank one) while the end-point Hamiltonian acquires the canonical Jordan-block form (i.e., it loses its diagonalizability). An optimal measure of the distance from the final catastrophe is finally found in a universal, exact formula for the spectrum of the metric.
An an exactly solvable quantum model is found to sample the evolution towards the sudden and complete loss of observability. For the purpose we choose an N-level system. While the time runs from 0 to 1, the process (leading to the collapse) is controlled by a toy-model Hamiltonian H and by a unitary-evolution guaranteeing minimally anisotropic (i.e., unique) Hilbert-space metric. The process of the degeneracy of the real N-plet of energy levels is studied without the usual assumption of adiabaticity. The initial Hamiltonian is diagonal, the initial metric is chosen as identity. Once the system reaches the observability horizon, the metric becomes singular (of rank one) while the end-point Hamiltonian acquires the canonical Jordan-block form (i.e., it loses its diagonalizability). An optimal measure of the distance from the final catastrophe is finally found in a universal, exact formula for the spectrum of the metric.
The bound-state spectrum of a Hamiltonian H is assumed real in a non-empty domain of physical values of parameters. This means that for these parameters, H may be called crypto-Hermitian, i.e. made Hermitian via an ad hoc choice of the inner product in the physical Hilbert space of quantum bound states (i.e. via an ad hoc construction of the operator Θ called the metric). The name quantum catastrophe is then assigned to the N -tuple-exceptional-point crossing, i.e. to the scenario in which we leave the domain along such a path that at the boundary of , an N -plet of bound-state energies degenerates and, subsequently, complexifies. At any fixed N ⩾ 2, this process is simulated via an N × N benchmark effective matrix Hamiltonian H . It is being assigned such a closed-form metric which is made unique via an N -extrapolation-friendliness requirement. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
To the existing list of alternative formulations of quantum mechanics, a new version of the non-Hermitian interaction picture is added. What is new is that, in contrast to the more conventional non-Hermitian model-building recipes, the primary information about the observable phenomena is provided not only by the Hamiltonian but also by an additional operator with a real spectrum (say, R(t)) representing another observable. In the language of physics, the information carried by R(t)≠R†(t) opens the possibility of reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a specific quantum phase transition. In parallel, the unitarity of the system remains guaranteed, as usual, via a time-dependent inner-product metric Θ(t). From the point of view of mathematics, the control of evolution is provided by a pair of conjugate Schrödiner equations. This opens the possibility od an innovative dyadic representation of pure states, by which the direct use of Θ(t) is made redundant. The implementation of the formalism is illustrated via a schematic cosmological toy model in which the canonical quantization leads to the necessity of working with two conjugate Wheeler-DeWitt equations. From the point of view of physics, the “kinematical input” operator R(t) may represent either the radius of a homogeneous and isotropic expanding empty Universe or, if you wish, its Hubble radius, or the scale factor a(t) emerging in the popular Lemaitre-Friedmann-Robertson-Walker classical solutions, with the exceptional-point singularity of the spectrum of R(t) mimicking the birth of the Universe (“Big Bang”) at t=0.
The description of unitary evolution using non-Hermitian but ‘hermitizable’ Hamiltonians H is feasible via an ad hoc metric Θ = Θ ( H ) and a (non-unique) amendment 〈 ψ 1 | ψ 2 〉 → 〈 ψ 1 | Θ | ψ 2 〉 of the inner product in Hilbert space. Via a proper fine-tuning of Θ ( H ) this opens the possibility of reaching the boundaries of stability (i.e. exceptional points) in many quantum systems sampled here by the fairly realistic Bose–Hubbard (BH) and discrete anharmonic oscillator (AO) models. In such a setting, it is conjectured that the EP singularity can play the role of a quantum phase-transition interface between different dynamical regimes. Three alternative ‘AO ↔ BH’ implementations of such an EP-mediated dynamical transmutation scenario are proposed and shown, at an arbitrary finite Hilbert-space dimension N , exact and non-numerical.
A unitary-evolution process leading to an ultimate collapse and to a complete loss of observability alias quantum phase transition is studied. A specific solvable N−state model is considered, characterized by a non-stationary non-Hermitian Hamiltonian. Our analysis uses quantum mechanics formulated in Schrödinger picture in which, in principle, only the knowledge of a complete set of observables (i.e., operators Λj) enables one to guarantee the uniqueness of the related physical Hilbert space (i.e., of its inner-product metric Θ). Nevertheless, for the sake of simplicity, we only assume the knowledge of just a single input observable (viz., of the energy-representing Hamiltonian H≡Λ1). Then, out of all of the eligible and Hamiltonian-dependent “Hermitizing” inner-product metrics Θ=Θ(H), we pick up just the simplest possible candidate. Naturally, this slightly restricts the scope of the theory, but in our present model, such a restriction is more than compensated for by the possibility of an alternative, phenomenologically better motivated constraint by which the time-dependence of the metric is required to be smooth. This opens a new model-building freedom which, in fact, enables us to force the system to reach the collapse, i.e., a genuine quantum catastrophe as a result of the mere conventional, strictly unitary evolution.
In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian HPU of the model is PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that HPU becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schr\"odinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.