arXiv subjects
Publications and source records attributed to Miloslav Znojil.
Via elementary examples it is demonstrated that the singularities of classical physics (sampled by the Big Bang in cosmology) need not necessarily get smeared out after quantization. It is proposed that the role of quantum singularities can be played by the so called Kato's exceptional-point spectral degeneracies.
Time and again, non-conventional forms of Lagrangians with non-quadratic velocity dependence have found attention in the literature. For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics including specifically the types of the Liénard class; for another, very often the problem of their quantization opens up multiple branches of the corresponding Hamiltonians, ending up with the presence of singularities in the associated eigenfunctions. In this article, we furnish a brief review of the classical theory of such Lagrangians and the associated branched Hamiltonians, starting with the example of Liénard-type systems. We then take up other cases where the Lagrangians depend upon the velocity with powers greater than two while still having a tractable mathematical structure, while also describing the associated branched Hamiltonians for such systems. For various examples, we emphasize upon the emergence of the notion of momentum-dependent mass in the theory of branched Hamiltonians.
Quantum mechanics of unitary systems is considered in quasi-Hermitian representation. In this framework the concept of perturbation is found counterintuitive, for three reasons. The first one is that in this formalism we are allowed to change the physical Hilbert-space norm. Thus, in a preselected Hamiltonian $H(λ)=H_0+λ\,H_1$ the size (and, hence, influence) of the perturbation cannot always be kept under a reliable control. Often, an enhanced sensitivity to perturbations is observed, for this reason, in open quantum systems. Second, even when we consider just a closed quantum system in which the influence of $H_1\neq H_1^\dagger$ is guaranteed to be small, the correct probabilistic interpretation of the system remains ambiguous, mainly due to the non-uniqueness of the physical Hilbert-space inner-product metric~$Θ$. Third, even if we decide to ignore the ambiguity and if we pick up just any one of the eligible metrics (which reduces the scope of the theory of course), such a choice would still vary with $λ$. In our paper it is shown that all of these three obstacles can be circumvented via just a mild amendment of the Rayleigh-Schrödinger perturbation-expansion approach. The flexibility of $Θ=Θ(λ)$ is shown to remain tractable while opening several new model-building horizons including the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive.
In quantum mechanics (formulated, say, in Schrödinger picture) only the knowledge of a complete set of observables $Λ_j$ enables us to declare the related physical inner product (i.e., the Hilbert-space metric $Θ$ such that $Λ_j^\dagger Θ=Θ\,Λ_j$, i.e., such that $Θ=Θ(Λ_j)$) unique. In many applications people simplify the model and consider just a single input observable (mostly an energy-representing Hamiltonian $Λ_1=H$) and pick up, out of all of the eligible metrics $Θ=Θ(H)$, just the simplest candidate (typically, in the case of the special self-adjoint input $H$ we virtually always work with trivial $Θ=I$). As long as this forces us to admit only the self-adjoint forms of any other input observable $Λ_j$, the scope of the theory is, without any truly meaningful phenomenological reason, restricted. In our present paper we describe a strictly non-numerical $N$ by $N$ matrix model in which such a restriction is replaced by another, phenomenologically non-equivalent restriction in which $Θ\neq I$ and in which the system reaches a collapse (i.e., a loss-of-bservability catastrophe) via unitary evolution.
Non-stationary version of unitary quantum mechanics formulated in non-Hermitian (or, more precisely, in hiddenly Hermitian) interaction-picture representation is illustrated via an elementary $N$ by $N$ matrix Hamiltonian $H(t)$ mimicking a 1D-box system with physics controlled by time-dependent boundary conditions. The model is presented as analytically solvable at $N=2$. Expressis verbis, this means that for both of the underlying Heisenbergian and Schrödingerian evolution equations the generators (i.e., in our notation, the respective operators $Σ(t)$ and $G(t)$) become available in closed form. Our key message is that contrary to the conventional beliefs and in spite of the unitarity of the evolution of the system, neither its "Heisenbergian Hamiltonian" $Σ(t)$ nor its "Schrödingerian Hamiltonian" $G(t)$ possesses a real spectrum or the conjugate pairs of complex eigenvalues. This means that neither one of these "Hamiltonians" can be pseudo-Hermitian alias PT-symmetric.
A new generalization of the Calogero's rational ($A_N$) many-body quantum model is proposed and studied. The key innovation lies in an asymmetrization of the Calogero's two-body interaction. In the generalized model the exact solvability is preserved while the Calogeero's single coupling constant $C$ is replaced by a certain multiplet of freely variable parameters. The resulting Calogero-like Hamiltonian ceases to be re-arrangement-invariant. As a consequence, the degeneracy of the bound-state spectrum gets partially unfolded.
In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state $ψ$ is controlled, in Hilbert space ${\cal L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces $\mathfrak{h}$ by its isospectral but non-Hermitian avatar $H = Ω^{-1}\mathfrak{h}Ω$ with $Ω^\daggerΩ= Θ\neq I$. Although acting in another, manifestly unphysical Hilbert space ${\cal H}$, the amended Hamiltonian $H \neq H^\dagger$ can be perceived as self-adjoint with respect to the amended inner-product metric $Θ$. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of $H$ we introduce and describe a specific new family of Hamiltonians $H$ for which the metrics $Θ$ become available in closed form.
A $(K+1)-$plet of non-Hermitian and time-dependent operators (say, $Λ_j(t)$, $j=0,1,\ldots,K$) can be interpreted as the set of observables characterizing a unitary quantum system. What is required is the existence of a self-adjoint and, in general, time-dependent operator (say, $Θ(t)$ called inner product metric) making the operators quasi-Hermitian, $Λ_j^\dagger(t)Θ(t)=Θ(t)Λ_j(t)$. The theory (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the states $ψ(t)$ (realized, via Schrödinger-type equation, by a generator, say, $G(t)$) and of the observables themselves (a different generator (say, $Σ(t)(t)$) occurs in the related non-Hermitian Heisenberg-type equation). Every $Λ_j(t)$ (and, in particular, Hamiltonian $H(t)=Λ_0(t)$) appears isospectral to its hypothetical isospectral and self-adjoint (but, by assumption, prohibitively user-unfriendly) avatar $λ_j(t)=Ω(t)Λ_j(t)Ω^{-1}(t)$ with $Ω^\dagger(t)Ω(t)=Θ(t)$. In our paper the key role played by identity $H(t)=G(t)+Σ(t)$ is shown to imply that there exist just three alternative meaningful implementations of the NIP approach, viz., ``number one'' (a ``dynamical'' strategy based on the knowledge of $H(t)$), ``number two'' (a ``kinematical'' one, based on the Coriolis force $Σ(t)$) and ``number three'' (in the literature, such a construction based on $G(t)$ is most popular but, paradoxically, it is also most complicated).
It is well known that the unitary evolution of a closed $M-$level quantum system can be generated by a non-Hermitian Hamiltonian $H$ with real spectrum. Its Hermiticity can be restored via an amended inner-product metric $Θ$. In Hermitian cases the evaluation of the spectrum (i.e., of the bound-state energies) is usually achieved by the diagonalization of the Hamiltonian. In the non-Hermitian (or, more precisely, in the $Θ-$quasi-Hermitian) quantum mechanics we conjecture that the role of the diagonalized-matrix solution of the quantum bound-state problem could be transferred to a maximally sparse ``zig-zag-matrix'' representation of the Hamiltonians.
In an amended version of non-Hermitian interaction picture we propose to work with the states $ψ(t)$ in a dyadic representation. The control of evolution via two conjugate Schrödiner equations then renders the usual necessity of the construction of the time-dependent inner-product-metric operator $Θ(t)$ redundant. The primary information about dynamics is assumed carried by a non-Hamiltonian observable (say, $R(t)$). A specific realization of phase transitions is then rendered possible via the Kato's exceptional-point (EP) degeneracy of the eigenvalues of $R(t)$ at the EP time $t=t^{(EP)}$. For illustration a cosmological model is proposed mimicking the unitary-evolution birth of the Universe from an initial quantum Big Bang singularity.
Quantum mechanics of closed, unitary quantum systems can be formulated in non-Hermitian interaction picture (NIP) in which both the states and the observables vary with time. Then, in general, not only the Schrödinger-equation generators $G(t)$ but also the Heisenberg-equation generators $Σ(t)$ are phenomenologically irrelevant, with spectra which are, in general, complex. Only the sum $H(t)=G(t)+Σ(t)$ retains the standard physical meaning of instantaneous energy. For illustration, the ``wrong-sign'' quartic oscillators are recalled and reconsidered.
The classical Coriolis force finds its quantum analogue in the difference $Σ(t)=H(t)-G(t)$ where the ``true'', observable Hamiltonian $H(t)$ represents the instantaneous energy. The other, ``false'' Hamiltonian $G(t)$ generates the time-evolution of wave functions. Whenever $Σ(t)\neq 0$, quantum mechanics acquires an interaction-picture form. Then, the time-evolution of every observable is generated by the Coriolis operator $Σ(t)$ ({\it alias} ``Heisenberg'' Hamiltonian) itself. In the paper a sequence of alternative formulae for $Σ(t)$ is derived under the assumption of an $N-$term factorization of the Dyson-map operator $Ω(t)$ (defined as converting a preselected quasi-Hermitian $H(t)$ into its conventional self-adjoint avatar). It is shown that in the resulting innovative formalism called ``factorization-based non-Hermitian interaction picture'' (FNIP) one has a choice between $N+1$ alternative forms of the description of quantum dynamics, one of which may prove, for the underlying quantum system, optimal. For illustration, the ``wrong-sign'' anharmonic oscillator model is recalled.
A reformulation of quantum mechanics called ``hybrid interaction picture'' (HIP) is proposed. In contrast to its current non-Hermitian-interaction-picture (NIP) predecessor the new formalism combines a simplification of the NIP Hilbert-space metric with an innovative preconditioning of the observables of interest. Via an illustrative example it is shown that the idea might open new ways towards optimal model-building strategies in quantum theory of unitary systems.
An elucidation of the current state of art in quasi-Hermitian quantum theory (QHQT) as inspired by the recent paper by Alase et al (J. Phys. A: Math. Theor. 55 (2022) 244003, paper [1]) is offered. We point out that the author's main discovery (viz., that the QHQT does not extend the standard quantum theory) is not new. In a related comment on the author's method of proof performed in ``the framework of general probabilistic theories'' (GPT) we add that also in this context a few other, mathematically consistent GPT-like theories are already available in the literature (pars pro toto we mention the results using the so called effect algebras). Thirdly, the ``intriguing open question'' about ``what possible constraints, if any, could lead to such a meaningful extension'' is given a tentative answer: The constraint could be just the generalized, non-stationary version of the quasi-Hermiticity.
In the recently quickly developing context of quantum mechanics of unitary systems using a time-independent non-Hermitian Hamiltonian $H$ (having real spectrum and defined as acting in an unphysical but user-friendly Hilbert space ${\cal R}_N^{(0)}$), the present paper introduces and describes a set of constructive returns of the description to one of the correct and eligible physical Hilbert spaces ${\cal R}_0^{(j)}$. The superscript $j$ may run from $j=0$ to $j=N$. In the $j=0$ extreme of the theory the construction is currently well known and involves solely the inner product metric $Θ=Θ(H)$. The Hamiltonian $H$ itself remains unchanged. At $j=N$ the inner-product metric remains trivial and only the Hamiltonian must be Hermitized, $H \to \mathfrak{h} = Ω\,H\,Ω^{-1}=\mathfrak{h}^\dagger$. At the remaining superscripts $j=1,2,\ldots, N-1$, a new, hybrid form of the construction of a consistent quantum model is proposed, requiring a simultaneous amendment of both the metric and the Hamiltonian. In applications, one of these options is expected to be optimal for a given $H$ in a way illustrated by a schematic three-state example.
In 1956, Freeman Dyson discovered that the practical solution of Schrödinger equation may be facilitated when one replaces the standard self-adjoint Hamiltonian $\mathfrak{h}=\mathfrak{h}^\dagger$ by its manifestly non-Hermitian isospectral avatar $H =Ω^{-1} \mathfrak{h} Ω$ with $Ω^\daggerΩ=Θ\neq I$. The standard unitary interpretation of the evolution can be then achieved either by the ``operator transformation'' (OT) reconstruction of $\mathfrak{h}$, or via a metric-amendment (MA) change of the Hilbert space, ${\cal H}_{auxiliary}\to {\cal H}_{physical}$. In our present letter we describe an alternative Hermitization of a ``hybrid form'' (HF) which simply combines the OT model-building strategy with the MA model-building strategy. The merits of the approach are illustrated using a schematic two-state quantum system.
A family of non-Hermitian but ${\cal PT}-$symmetric $2J$ by $2J$ toy-model tridiagonal-matrix Hamiltonians $H^{(2J)}=H^{(2J)}(t)$ with $J=K+M=1,2,\ldots$ and $t<J^2$ is studied, for which a real but non-Hermitian $2K$ by $2K$ tridiagonal-submatrix component $C(t)$ of the Hamiltonian is assumed coupled to its other two complex but Hermitian $M$ by $M$ tridiagonal-submatrix components $A(t)$ and $B(t)$. By construction, (i) all of the submatrices get decoupled at $t=t_M=M\,(2J-M)$ with $M=1,2,\ldots,J$; (ii) at all of the parameters $t=t_M$ with $M=J-K=0,1,\ldots,J-1$ the Hamiltonian ceases to be diagonalizable exhibiting the Kato's exceptional-point degeneracy of order $2K$; (iv) the system's ${\cal PT}-$symmetry gets spontaneously broken when $t\leq t_{J-1}=J^2-1$.
For the displaced harmonic double-well oscillator the existence of exact polynomial bound states at certain displacements $d\,$ is revealed. The $N-$plets of these quasi-exactly solvable (QES) states are constructed in closed form. For non-QES states, Schrödinger equation can still be considered ``non-polynomially exactly solvable'' (NES) because the exact left and right parts of the wave function (proportional to confluent hypergeometric function) just have to be matched in the origin.