The phenomenon of degeneracy of an N-plet of bound states is studied in the framework of the quasi-Hermitian (a.k.a. PT-symmetric) formulation of quantum theory of closed systems. For a general non-Hermitian Hamiltonian H=H(λ) such a degeneracy may occur at a real Kato's exceptional point λ^{(EPN)} of order N and of the geometric multiplicity alias clusterization index K. The corresponding unitary process of collapse (loss of observability) can be then interpreted as a generic quantum phase transition. The dedicated literature deals, predominantly, with the non-numerical benchmark models of the simplest processes where K=1. In our present paper it is shown that in the "anomalous" dynamical scenarios with 1<K≤N/2 an analogous approach is applicable. A multiparametric anharmonic-oscillator-type exemplification of such systems is constructed as a set of real-matrix N by N Hamiltonians which are exactly solvable, maximally non-Hermitian, and labeled by specific ad hoc partitionings R(N) of N.
The phenomenon of degeneracy of an N-plet of bound states is studied in the framework of the quasi-Hermitian (a.k.a. PT-symmetric) formulation of quantum theory of closed systems. For a general non-Hermitian Hamiltonian H=H(λ) such a degeneracy may occur at a real Kato's exceptional point λ^{(EPN)} of order N and of the geometric multiplicity alias clusterization index K. The corresponding unitary process of collapse (loss of observability) can be then interpreted as a generic quantum phase transition. The dedicated literature deals, predominantly, with the non-numerical benchmark models of the simplest processes where K=1. In our present paper it is shown that in the "anomalous" dynamical scenarios with 1<K≤N/2 an analogous approach is applicable. A multiparametric anharmonic-oscillator-type exemplification of such systems is constructed as a set of real-matrix N by N Hamiltonians which are exactly solvable, maximally non-Hermitian, and labeled by specific ad hoc partitionings R(N) of N.
The phenomenon of degeneracy of an N−plet of bound states is studied in the framework of quantum theory of closed (i.e., unitary) systems. For an underlying Hamiltonian H=H(λ) the degeneracy occurs at a Kato's exceptional point λ(EPN) of order N and of the spectral geometric multiplicity K2 and K>1 benchmark models of the process of degeneracy becomes feasible and non-numerical for a broad class of specific, maximally non-Hermitian anharmonic-oscillator toy-model Hamiltonians. An exhaustive classification of non-equivalent processes is given by a partitioning of the unperturbed spectrum into equidistant and centered unperturbed subspectra.
Via several toy-model quantum Hamiltonians H(λ) of a non-tridiagonal low-dimensional matrix form the existence of unusual observability horizons is revealed. At the corresponding limiting values of parameter λ=λ(critical) these new types of quantum phase transitions are interpreted as the points of confluence of several decoupled Kato's exceptional points of equal or different orders. Such a phenomenon of degeneracy of non-Hermitian degeneracies seems to ask for a reclassification of the possible topologies of the complex energy Riemann surfaces in the vicinity of branch points.
Quantum phase transition is interpreted as an evolution, at the end of which a parameter-dependent Hamiltonian H(g) loses its observability. In the language of mathematics, such a quantum catastrophe occurs at an exceptional point of order N (EPN). Although the Hamiltonian H(g) itself becomes unphysical in the limit of g→gEPN, it is shown that it can play the role of an unperturbed operator in a perturbation-approximation analysis of the vicinity of the EPN singularity. Such an analysis is elementary at N≤3 and numerical at N≥5, so we pick up N=4. We demonstrate that the specific EP4 degeneracy becomes accessible via a unitary evolution process realizable inside a parametric domain Dphysical, the boundaries of which are determined non-numerically. Possible relevance of such a mathematical result in the context of non-Hermitian photonics is emphasized.
A family of non-Hermitian real and tridiagonal-matrix candidates H(N)(λ)=H0(N)+λW(N)(λ) for a hiddenly Hermitian (a.k.a. quasi-Hermitian) quantum Hamiltonian is proposed and studied. Fairly weak assumptions are imposed upon the unperturbed matrix [the square-well-simulating spectrum of H0(N) is not assumed equidistant)] and upon its maximally non-Hermitian N-parametric antisymmetric-matrix perturbations [matrix W(N)(λ) is not even required to be PT-symmetric]. Despite that, the “physical” parametric domain D[N] is (constructively) shown to exist, guaranteeing that in its interior, the spectrum remains real and non-degenerate, rendering the quantum evolution unitary. Among the non-Hermitian degeneracies occurring at the boundary ∂D[N] of the domain of stability, our main attention is paid to their extreme version corresponding to Kato’s exceptional point of order N (EPN). The localization of the EPNs and, in their vicinity, of the quantum-phase-transition boundaries ∂D[N] is found feasible, at the not too large N, using computer-assisted symbolic manipulations, including, in particular, the Gröbner-basis elimination and the high-precision arithmetics.
Up to these days, the popular PT-symmetric imaginary cubic oscillator did not find any consistent probabilistic quantum-mechanical interpretation because its Hamiltonian has been shown, by mathematicians, intrinsic-exceptional-point (IEP) singular. In the paper we explain why there is even no reasonable small-perturbation-based regularization of the similar unacceptable (i.e., IEP-singular) quantum models. The explanation is based on a partial formal analogy of the IEP singularity with the conventional exceptional point (EP). What is important is that we are able to construct a simplified N by N-matrix (and exactly solvable) toy-model Hamiltonian admitting the asymptotic (i.e., high-excitation) EP-related wave-function degeneracy which, in some sense (i.e., in the limit of large N) mimics several aspects of its IEP analogue. In this comparison, the difference is that the regularization of the EP singularities is possible (using an ad hoc perturbation of size O(1/N)) while an analogous regularization of the IEP singularity is not (we have to consider N→∞).
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.