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 $H$ .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 $H$.

quant-pharXiv:2511.20404Nov 26, 20250 citations

Microscopic observation of magnon bound states and their dynamics

Takeshi Fukuhara, Peter Schauß, Manuel Endres +4 more

arXiv:1305.6598NatureSep 24, 2013459 citations

Algebraic analysis of physical and spurious states in Dyson boson mapping

P. Park

Dyson boson mapping of a system of 2n fermions distributed among k single-particle states uses a space generated by two-index ideal bosons. We define a unitary group U(k) to classify this ideal-boson space. The physical subspace is shown to correspond to the antisymmetric irreducible representation [ ${1}^{2n}$ ]. This classification enables one to introduce a ``Majorana'' operator S which is a linear combination of one- and two-body Casimir operators of U(k). A zero eigenvalue of S characterizes the physical subspace while positive eigenvalues identify all other irreducible representations that occur. The Hermitian boson image of the Hamiltonian does not connect physical and spurious states because it is a function of the generators of U(k). All the nonphysical eigenvalues and eigenstates of a physical boson Hamiltonian (Dyson boson image of a physical fermion Hamiltonian) which could be Hermitian or non-Hermitian, can be removed without affecting the physical eigenvalues and eigenvectors, by adding a suitable multiple of S to the Hamiltonian.Dyson boson mapping of a system of 2n fermions distributed among k single-particle states uses a space generated by two-index ideal bosons. We define a unitary group U(k) to classify this ideal-boson space. The physical subspace is shown to correspond to the antisymmetric irreducible representation [${1}^{2n}$]. This classification enables one to introduce a ``Majorana'' operator S which is a linear combination of one- and two-body Casimir operators of U(k). A zero eigenvalue of S characterizes the physical subspace while positive eigenvalues identify all other irreducible representations that occur. The Hermitian boson image of the Hamiltonian does not connect physical and spurious states because it is a function of the generators of U(k). All the nonphysical eigenvalues and eigenstates of a physical boson Hamiltonian (Dyson boson image of a physical fermion Hamiltonian) which could be Hermitian or non-Hermitian, can be removed without affecting the physical eigenvalues and eigenvectors, by adding a suitable multiple of S to the Hamiltonian.

Physical Review CFeb 1, 198724 citations

Three magnon bound states

Chanchal K. Majumdar, Goutam Mukhopadhyay

Physics Letters AMar 1, 19707 citations

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

{1}^{2n}

]. This classification enables one to introduce a ``Majorana'' operator S which is a linear combination of one- and two-body Casimir operators of U(k). A zero eigenvalue of S characterizes the physical subspace while positive eigenvalues identify all other irreducible representations that occur. The Hermitian boson image of the Hamiltonian does not connect physical and spurious states because it is a function of the generators of U(k). All the nonphysical eigenvalues and eigenstates of a physical boson Hamiltonian (Dyson boson image of a physical fermion Hamiltonian) which could be Hermitian or non-Hermitian, can be removed without affecting the physical eigenvalues and eigenvectors, by adding a suitable multiple of S to the Hamiltonian.