arXiv subjects

Miloslav Znojil

Publications and source records attributed to Miloslav Znojil.

At least 55 records · Page 3Linked to original sources

arXiv2022

Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology

Recently, Wang with Stankiewicz (Phys. Lett. B 800 (2020) 135106) opposed the widespread belief that due to quantization, the Big Bang singularity {\em must} necessarily get smeared and replaced by a non-singular "Big Bounce" process. Their claim is supported in a methodically independent manner in which just a highly schematic model (viz., the discrete and drastically simplified Wheeler-DeWitt equation) is considered. Our comparatively elementary argumentation is constructive, based on the assumption that the evolution of the Universe can fully consistently be described using an innovative, hiddenly Hermitian version of quantum theory formulated in a properly upgraded Dirac's interaction picture.

gr-qc↗

arXiv2022

Factorized Hilbert-space metrics and non-commutative quasi-Hermitian observables

It is well known that an (in general, non-commutative) set of non-Hermitian operators $Λ_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus the underlying physical Hilbert-space metric $Θ$ are all represented in terms of an auxiliary operator $(N+1)-$plet $Z_k$, $k=0,1,\ldots,N$. Our formalism degenerates to the ${\cal PT}-$symmetric quantum mechanics at $N=2$, with metric $Θ=Z_2Z_1$, parity ${\cal P}=Z_2$, charge ${\cal C}=Z_1$ and Hamiltonian $H=Z_0$.

quant-ph↗

arXiv2022

Arnold's potentials and quantum catastrophes II

The well known phenomenon of avoided level crossing (ALC) can be perceived as a quantum analogue of the Thom's catastrophes in classical dynamical systems. One-dimensional Schrödinger equation is chosen for illustration. In constructive manner, a family of confining polynomial potentials is considered, characterized by the presence of an $N-$plet of high barriers separating the $(N+1)-$plet of deep valleys. The bifurcations of the long-time classical equilibria are shown paralleled by the ALCs in the quantum low-lying spectra. Every tunneling-controlled fine-tuned switch of dominance between the valleys is finally interpreted as a change of the topological structure of the probability density representing a genuine quantum relocalization catastrophe.

quant-ph↗

arXiv2022

Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians

In the popular ${\cal PT}-$symmetry-based formulation of quantum mechanics of closed systems one can build unitary models using non-Hermitian Hamiltonians (i.e., $H \neq H^\dagger$) which are Hermitizable (so that one can write, simultaneously, $H = H^\ddagger$). The essence of the trick is that the reference Hilbert space $\cal R$ (in which we use the conventional inner product $\langle ψ_a|ψ_b\rangle$ and write $H \neq H^\dagger$) is declared unphysical. The necessary Hermiticity of the Hamiltonian $H = H^\ddagger$ can be then achieved by the mere metric-mediated amendment $\langle ψ_a|Θ|ψ_b\rangle$ to the inner product. This converts $\cal R$ into a correct physical Hilbert space $\cal H$. The feasibility of the construction is based on a factorization postulate $Θ={\cal PC}$ where, usually, ${\cal P}$ is parity and ${\cal C}$ is charge. In our paper we propose a more general factorization recipe in which one constructs $Θ=Z_NZ_{N-1}\ldots Z_1$, at any $N$, in terms of suitable auxiliary pre-metric operators $Z_k$.

quant-ph↗

arXiv2022

Confluences of exceptional points and a systematic classification of quantum catastrophes

Specific quantum phase transitions of our interest are assumed associated with the fall of a closed, unitary quantum system into its exceptional-point (EP) singularity. The physical realization of such a "quantum catastrophe" (connected, typically, with an instantaneous loss of the diagonalizability of the corresponding parameter-dependent Hamiltonian $H(g)$) depends, naturally, on the formal mathematical characteristics of the EP, i.e., in essence, on its so called algebraic multiplicity $N$ and geometric multiplicity $K$. In our paper we assume that both of them are finite, and we illustrate and discuss, using several solvable toy models, some of the most elementary mechanisms of the EP-merger realization of the process of the transition $g \to g^{(EP)}$.

quant-ph↗

arXiv2021

Which Metrics Are Consistent with a Given Pseudo-Hermitian Matrix?

Given a diagonalizable $N\times N$ matrix $H$, whose non-degenerate spectrum consists of $p$ pairs of complex conjugate eigenvalues and additional $N-2p$ real eigenvalues, we determine all metrics $M$, of all possible signatures, with respect to which $H$ is pseudo-hermitian. In particular, we show that any compatible $M$ must have $p$ pairs of opposite eigenvalues in its spectrum so that $p$ is the minimal number of both positive and negative eigenvalues of $M$. We provide explicit parametrization of the space of all admissible metrics and show that it is topologically a $p$-dimensional torus tensored with an appropriate power of the group $Z_2$.

math-ph↗

arXiv2021

Quantum mechanics using two auxiliary inner products

The current applications of non-Hermitian but ${\cal PT}-$symmetric Hamiltonians $H$ cover several, mutually not too closely connected subdomains of quantum physics. Mathematically, the split between the open and closed systems can be characterized by the respective triviality and non-triviality of an auxiliary inner-product metric $Θ=Θ(H)$. With our attention restricted to the latter, mathematically more interesting unitary-evolution case we show that the intuitive but technically decisive simplification of the theory achieved via an "additional" ${\cal PCT}-$symmetry constraint upon $H$ can be given a deeper mathematical meaning via introduction of a certain second auxiliary inner product.

quant-ph↗

arXiv2021

Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via ${\cal PT}-$symmetric and exactly solvable linear-Bose-Hubbard building blocks

It is well known that using the conventional non-Hermitian but ${\cal PT}-$symmetric Bose-Hubbard Hamiltonian with real spectrum one can realize the Bose-Einstein condensation (BEC) process in an exceptional-point limit of order $N$. Such an exactly solvable simulation of the BEC-type phase transition is, unfortunately, incomplete because the standard version of the model only offers an extreme form of the limit characterized by a minimal geometric multiplicity $K=1$. In our paper we describe a rescaled and partitioned direct-sum modification of the linear version of the Bose-Hubbard model which remains exactly solvable while admitting any value of $K\geq 1$. It offers a complete menu of benchmark models numbered by a specific combinatorial scheme. In this manner, an exhaustive classification of the general BEC patterns with any geometric multiplicity is obtained and realized in terms of an exactly solvable generalized Bose-Hubbard model.

quant-ph↗

arXiv2021

Avoided level crossings in quasi-exact approach

The existence of quantum tunneling opens the possibility of a sudden spatial relocalization of a system after a minor modification of its parameters. Such a quantum analogue of the Thom's classical catastrophe would manifest itself, experimentally, via a reordering of the maxima of the probability density paralleled by avoided crossings of the energy levels. Any model (described, say, by an analytic potential with several pronounced minima) is difficult to describe near such an instability because the phenomenon is oversensitive to perturbations. A systematic exact (or, better, quasi-exact) construction of the relocalization instants is proposed here. Its application is considered in the one-dimensional critical-instant setup. The approach is shown to yield the mutually consistent non-polynomial analytic potentials together with the related non-numerical multi-Gaussian-shaped wave functions.

quant-ph↗

arXiv2021

Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians

A phenomenological Hamiltonian of a closed (i.e., unitary) quantum system is assumed to have an $N$ by $N$ real-matrix form composed of a unperturbed diagonal-matrix part $H^{(N)}_0$ and of a tridiagonal-matrix perturbation $λ\,W^{(N)}(λ)$. The requirement of the unitarity of the evolution of the system (i.e., of the diagonalizability and of the reality of the spectrum) restricts, naturally, the variability of the matrix elements to a "physical" domain ${\cal D}^{[N]} \subset \mathbb{R}^d$. We fix the unperturbed matrix (simulating a non-equidistant, square-well-type unperturbed spectrum) and we only admit the maximally non-Hermitian antisymmetric-matrix perturbations. This yields the hiddenly Hermitian model with the measure of perturbation $λ$ and with the $d=N$ matrix elements which are, inside ${\cal D}^{[N]}$, freely variable. Our aim is to describe the quantum phase-transition boundary $\partial {\cal D}^{[N]}$ (alias exceptional-point boundary) at which the unitarity of the system is lost. Our main attention is paid to the strong-coupling extremes of stability, i.e., to the Kato's exceptional points of order $N$ (EPN) and to the (sharply spiked) shape of the boundary $\partial {\cal D}^{[N]}$ in their vicinity. The feasibility of our constructions is based on the use of the high-precision arithmetics in combination with the computer-assisted symbolic manipulations (including, in particular, the Gröbner basis elimination technique).

math-ph↗

arXiv2021

Quantum phase transitions in nonhermitian harmonic oscillator

The Stone theorem requires that in a physical Hilbert space ${\cal H}$ 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 ${\cal K}$ in which $H$ is nonhermitian but ${\cal PT}-$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 ${\cal H}$. For a ${\cal PT}-$symmetric version of the spiked harmonic oscillator we show that in the dynamical regime of the unavoided level crossings such a reconstruction of ${\cal H}$ becomes feasible and, moreover, obtainable by non-numerical means. The general form of such a reconstruction of ${\cal H}$ enables one to render every exceptional unavoided-crossing point tractable as a genuine, phenomenologically most appealing quantum-phase-transition instant.

quant-ph↗

arXiv2021

Quantum phase transitions mediated by clustered non-Hermitian degeneracies

A broad family of phase transitions in the closed as well as open quantum systems is known to be mediated by a non-Hermitian degeneracy (a.k.a. exceptional point, EP) of the Hamiltonian. In the EP limit, in general, the merger of an $N-$plet of the energy eigenvalues is accompanied by a parallel (though not necessarily complete) degeneracy of eigenstates (forming an EP-asociated $K-$plet; in mathematics, $K$ is called the geometric multiplicity of the EP). In the literature, unfortunately, only the benchmark matrix models with $K=1$ can be found. In our paper the gap is filled: the EP-mediated quantum phase transitions with $K>1$ are called "clustered", and a family of benchmark models admitting such a clustering phenomenon is proposed and described. For the sake of maximal simplicity our attention is restricted to the real perturbed-harmonic-oscillator-type N by N matrix Hamiltonians which are exactly solvable and in which the perturbation is multiparametric (i.e., maximally variable) and antisymmetric (i.e., maximally non-Hermitian). A labeling (i.e., an exhaustive classification) of these models is provided by a specific partitioning of N.

quant-ph↗

arXiv2021

Paths of unitary access to exceptional points

During the early history of unitary quantum theory the Kato's exceptional points (EPs, a.k.a. non-Hermitian degeneracies) of Hamiltonians $H(λ)$ did not play any significant role, mainly due to the Stone theorem which firmly connected the unitarity with the Hermiticity. During the recent wave of optimism people started believing that the corridors of a unitary access to the EPs could be opened leading, say, to a new picture of quantum phase transitions via an {\it ad hoc} weakening of the Hermiticty (replaced by the quasi-Hermiticity). Subsequently, the pessimism prevailed (the paths of access appeared to be fragile). In a way restricted to the quantum physics of closed systems a return to optimism is advocated here: the apparent fragility of the corridors is claimed to follow from a misinterpretation of the theory in its quasi-Hermitian formulation. Several perturbed versions of the realistic many-body Bose-Hubbard model are chosen for illustration purposes.

quant-ph↗

arXiv2020

Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities

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 $K 2$ 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.

quant-ph↗

arXiv2020

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

Non-Hermitian but ${\cal PT}-$symmetric quantum system of an $N-$plet of bosons described by the three-parametric Bose-Hubbard Hamiltonian $H(γ,v,c)$ is picked up, in its special exceptional-point limit $c \to 0$ and $γ\to v$, as an unperturbed part of the family of generalized Bose-Hubbard-like Hamiltonians $\mathfrak{H}(λ)=H(v,v,0)+λ\,{\cal V}$ for which the unitarity of the perturbed system is required. This leads to the construction of two different families of Hamiltonians $\mathfrak{H}(λ)$. In the first one the number $N$ of bosons is assumed conserved while in the second family such an assumption is relaxed. In both cases the anisotropy of the related physical Hilbert space is shown reflected by a highly counterintuitive but operationally realizable structure of admissible perturbations $λ\,{\cal V}$.

math-ph↗

arXiv2020

Perturbation theory near degenerate exceptional points

In an overall framework of quantum mechanics of unitary systems a rather sophisticated new version of perturbation theory is developed. What is assumed is, firstly, that the perturbed Hamiltonians $H=H_0+λV$ are non-Hermitian and lie close to their unobservable exceptional-point (EP) degeneracy limit $H_0$. Secondly, in this EP limit, the geometric multiplicity $L$ of the degenerate unperturbed eigenvalue $E_0$ is assumed, in contrast to the majority of existing studies, larger than one. Under these assumptions the method of construction of the bound states is described. Its specific subtleties are illustrated via the leading-order recipe. The emergence of a counterintuitive connection between the value of $L$, the structure of the matrix elements of perturbations, and the possible loss of the stability and unitarity of the processes of the unfolding of the EP singularity is given a detailed explanation.

math-ph↗

arXiv2020

Anomalous mechanisms of the loss of observability in non-Hermitian quantum models

Quantum phase transitions in certain non-Hermitian systems controlled by non-tridiagonal Hamiltonian matrices are found anomalous. In contrast to the known models with tridiagonal-matrix structure in which the geometric multiplicity of the completely degenerate energy eigenvalue appears always equal to one, this multiplicity is found larger than one in the present models. The phenomenon is interpreted as a confluence of several decoupled Kato's exceptional points of equal or different orders.

quant-ph↗

arXiv2020

Relocalization switch in a triple quantum dot molecule in 2D

Tunneling between pronounced minima of a polynomial potential (simulating coupled quantum dots) can cause an abrupt quantum-catastrophic relocalization of density $|ψ(x,y, \ldots)|$. Non-numerical illustration is constructed, in 2D, using a non-separable sextic-oscillator model.

cond-mat.mes-hall↗