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arXiv:2511.20404

Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics

Aritra Ghosh, Adam Miranowicz, Miloslav Znojil

quant-phSubmitted Nov 25, 2025• Announced Nov 26, 2025• 0 citations• Revised Jan 15, 2026 (v2)
v1 · Nov 25, 2025v2 · Jan 15, 2026
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Abstract

In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates $|ψ_n \rangle$ of a Hamiltonian operator $H$ with real spectrum which is non-Hermitian, $H \neq H^\dagger$. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of $H$ into its isospectral Hermitian ``twin" $\mathfrak{h}= \mathfrak{h}^\dagger$ via one of the so-called Dyson maps $Ω: H \to \mathfrak{h}$. Naturally, the well known ambiguity of these $H-$dependent Dyson-map transformations implies also an ambiguity of the physical, $Ω-$dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible $H-$dependent Dyson maps $Ω=Ω(H)$ is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected $H$.

Journal reference
Symmetry-Basel 18 (2026) 189
DOI
10.3390/sym18010189
Comments
21 pp., written in honor of our friend, professor Jean-Pierre Gazeau, on the occasion of his 80th birthday

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In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates ∣ψn⟩|ψ_n \rangle∣ψn​⟩ of a Hamiltonian operator HHH with real spectrum which is non-Hermitian, H≠H†H \neq H^\daggerH=H†. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of HHH into its isospectral Hermitian ``twin" h=h†\mathfrak{h}= \mathfrak{h}^\daggerh=h† via one of the so-called Dyson maps Ω:H→hΩ: H \to \mathfrak{h}Ω:H→h. Naturally, the well known ambiguity of these H−H-H−dependent Dyson-map transformations implies also an ambiguity of the physical, Ω−Ω-Ω−dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible H−H-H−dependent Dyson maps Ω=Ω(H)Ω=Ω(H)Ω=Ω(H) is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected HHH.
We establish a quantitative adiabatic estimate for a class of finite-dimensional non-Hermitian Schrödinger dynamics. The original non-Hermitian Hamiltonian is assumed to be diagonalizable with real spectrum and non-crossing eigenvalues. We then construct a dynamically compatible time-dependent metric operator, and its positive square root defines a Dyson map. The associated Dyson-transformed Hamiltonian is Hermitian. When this Dyson-transformed Hamiltonian satisfies the Hermitian adiabatic assumption, the standard resolvent-projection method gives an explicit adiabatic estimate in the Hermitian representation. Pulling this estimate back through the Dyson map gives an approximation in the original non-Hermitian representation. The resulting Dyson-pulled-back projections are then compared with the spectral projections of the original non-Hermitian Hamiltonian by using a contour-resolvent estimate. The final bound contains two contributions: the pulled-back Hermitian adiabatic error and the projection-comparison error. A two-level non-Hermitian model is presented to illustrate the hypotheses of the theorem and the two contributions appearing in the final estimate.
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.
A recent surge of publications about non-Hermitian Hamiltonians has led to considerable controversy and — in our opinion — to some misunderstandings of basic quantum mechanics. The present paper scrutinizes the metric associated with a quasi-Hermitian Hamiltonian and its physical implications. The consequences of the non-uniqueness such as the question of the probability interpretation and the possible and forbidden choices of additional observables are investigated and exemplified by specific illustrative examples. In particular, it is argued that the improper identification of observables lies at the origin of the claimed violation of the brachistchrone transition time between orthogonal states. The need for further physical input to remove ambiguities is pointed out.PACS Nos.: 03.65.–w, 03.65.Ca, 03.65.Ta, 03.65.Xp
Abstract We propose and explore a scheme that leads to an infinite series of time-dependent Dyson maps which associate different Hermitian Hamiltonians to a uniquely specified time-dependent non-Hermitian Hamiltonian. We identify the underlying symmetries responsible for this feature respected by various Lewis–Riesenfeld invariants. The latter are used to facilitate the explicit construction of the Dyson maps and metric operators. As a concrete example for which the scheme is worked out in detail we present a two-dimensional system of oscillators that are coupled to each other in a non-Hermitian PT -symmetrical fashion.
In the framework of quasi-Hermitian quantum mechanics, the eligible operators of observables may be non-Hermitian, Aj≠Aj†, j=1,2,…,K. In principle, the standard probabilistic interpretation of the theory can be re-established via a reconstruction of physical inner-product metric Θ≠I, guaranteeing the quasi-Hermiticity Aj†Θ=ΘAj. The task is easy at K=1 because there are many eligible metrics Θ=Θ(A1). In our paper, the next case with K=2 is analyzed. The criteria of the existence of a shared metric, Θ=Θ(A1,A2), are presented and discussed.
Abstract I will provide a pedagogical introduction to non-Hermitian quantum systems that are PT-symmetric, that is they are left invariant under a simultaneous parity transformation (P) and time-reversal (T). I will explain how generalised versions of this antilinear symmetry can be utilised to explain that these type of systems possess real eigenvalue spectra in parts of their parameter spaces and how to set up a consistent quantum mechanical framework for them that enables a unitary time-evolution. In the second part I will explain how to extend this framework to explicitly time-dependent Hamiltonian systems and report in particular on recent progress made in this context. I will explain how to construct the essential key quantity in this framework, the time-dependent Dyson map and metric and solutions to the time-dependent Schrödinger equation, in an algebraic fashion, using time-dependent Darboux transformations, utilising Lewis-Riesenfeld invariants, point transformations and some approximation methods. I comment on the ambiguities of this metric and demonstrate that this can even lead to infinite series of metric operators. I conclude with some applications to PT-symmetrically coupled oscillators, demonstrate the equivalence of the time-dependent double wells and unstable anharmonic oscillators and show how the unphysical PT-symmetrically broken regions in the parameter space for the time-independent theory becomes physical in the explicitly time-dependent systems. I discuss how this leads to a prolongation of the otherwise rapidly decaying von Neumann entropy. The so-called sudden death of the entropy is stopped at a finite value.1