In quite a few recent quantum models one is allowed to make a given Hamiltonian H self-adjoint only after an ad hoc generalization of Hermitian conjugation, H†→H‡:= Θ −1H†Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠ I the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of H) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.In quite a few recent quantum models one is allowed to make a given Hamiltonian H self-adjoint only after an ad hoc generalization of Hermitian conjugation, H†→H‡:= Θ −1H†Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠ I the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of H) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.

Aug 14, 20161 citation

Quasi-hermitian quantum mechanics and a new class of user-friendly matrix hamiltonians

Olaf Lechtenfeld, Miloslav Znojil

Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA ${\mathcal{L}}$ ?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA ${\mathcal{H}}$ ?> H , the amended Hamiltonian H ≠ H † 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.Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA ${\mathcal{L}}$?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA ${\mathcal{H}}$?> H , the amended Hamiltonian H ≠ H † 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.

Journal of Physics Conference SeriesDec 1, 20230 citations

Quasi-Hermitian quantum mechanics and a new class of user-friendly matrix Hamiltonians

Olaf Lechtenfeld, Miloslav Znojil

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.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.

arXiv:2308.15043Aug 29, 20230 citations

Scattering theory with localized non-Hermiticities

Miloslav Znojil

In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that the well-known variability of the ad hoc choice of the metric $\ensuremath{\Theta}$ which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable $\ensuremath{\Theta}$, a fully plausible physical picture of the scattering can be recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that the well-known variability of the ad hoc choice of the metric $\ensuremath{\Theta}$ which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable $\ensuremath{\Theta}$, a fully plausible physical picture of the scattering can be recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.

arXiv:0805.2800Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJul 22, 200865 citations

Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models

Miloslav Znojil

Abstract In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that <?CDATA $H^\dagger \Theta = \Theta\,H$ ?> H † Θ = Θ H , the exactly solvable M-level bound-state models with arbitrary <?CDATA $M\leqslant \infty$ ?> M ⩽ ∞ are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just <?CDATA $(2M-1)$ ?> ( 2 M − 1 ) -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.Abstract In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that <?CDATA $H^\dagger \Theta = \Theta\,H$?> H † Θ = Θ H , the exactly solvable M-level bound-state models with arbitrary <?CDATA $M\leqslant \infty$?> M ⩽ ∞ are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just <?CDATA $(2M-1)$?> ( 2 M − 1 ) -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.

arXiv:2307.03439Journal of Physics A Mathematical and TheoreticalJul 19, 20232 citations

Hiddenly Hermitian quantum models: The concept of perturbations.

Miloslav Znojil

In conventional Schrodinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\dagger$ induces the change ${\cal L} \to {\cal K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\neq H^\dagger$ . In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into the third, correct physical Hilbert space ${\cal H}$ , unitarily equivalent to ${\cal L}$ . The situation encountered, typically, in ${\cal PT}-$ symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.In conventional Schrodinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\dagger$ induces the change ${\cal L} \to {\cal K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\neq H^\dagger$. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into the third, correct physical Hilbert space ${\cal H}$, unitarily equivalent to ${\cal L}$. The situation encountered, typically, in ${\cal PT}-$symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.

Aug 8, 20190 citations

Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA

{\mathcal{L}}

?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA

{\mathcal{H}}

?> H , the amended Hamiltonian H ≠ H † 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.

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.

Abstract In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that <?CDATA

H^\dagger \Theta = \Theta\,H

?> H † Θ = Θ H , the exactly solvable M-level bound-state models with arbitrary <?CDATA

M\leqslant \infty

?> M ⩽ ∞ are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just <?CDATA

(2M-1)

?> ( 2 M − 1 ) -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.

\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\dagger

induces the change

{\cal L} \to {\cal K}

of the Hilbert space of states, reflected by the loss of the Hermiticity of

H\neq H^\dagger

. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts

{\cal K}

into the third, correct physical Hilbert space

{\cal H}

, unitarily equivalent to

{\cal L}

. The situation encountered, typically, in

{\cal PT}-

symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.