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Two sample constructions of all the Hilbert spaces for a given operator $H$ of an observable

Miloslav Znojil

Published Jun 26, 2008• 0 citations
OpenAlex

Abstract

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\dagger$ with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states ${\cal H}^{(physical)}$. The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators $\Theta$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $\lambda \in (-1,1)$ in matrix $H^{(N)}(\lambda)$, the complete $N-$parametric set of metrics $\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)$ is recurrently defined by closed formula.

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Two sample constructions of all the Hilbert spaces for a given operator HHH of an observable
An elementary set of non-Hermitian NNN by NNN matrices H(N)(λ)≠[H(N)(λ)]†H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\daggerH(N)(λ)=[H(N)(λ)]† with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states H(physical){\cal H}^{(physical)}H(physical). The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators Θ\ThetaΘ) is addressed. The problem is shown exactly solvable and, for every size N=2,4,...N=2,4,...N=2,4,... and parameter λ∈(−1,1)\lambda \in (-1,1)λ∈(−1,1) in matrix H(N)(λ)H^{(N)}(\lambda)H(N)(λ), the complete N−N-N−parametric set of metrics Θα1,α2,...,αN(N)(λ)\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)Θα1​,α2​,...,αN​(N)​(λ) is recurrently defined by closed formula.
The complete menu of eligible metrics for a family of toy Hamiltonians H≠H†H \neq H^\daggerH=H† with real spectra
An elementary set of non-Hermitian NNN by NNN matrices H(N)(λ)≠[H(N)(λ)]†H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\daggerH(N)(λ)=[H(N)(λ)]† with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states H(physical){\cal H}^{(physical)}H(physical). The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators ΘΘΘ) is addressed. The problem is shown exactly solvable and, for every size N=2,4,...N=2,4,...N=2,4,... and parameter λ∈(−1,1)λ\in (-1,1)λ∈(−1,1) in matrix H(N)(λ)H^{(N)}(λ)H(N)(λ), the complete N−N-N−parametric set of metrics Θα1,α2,...,αN(N)(λ)Θ^{(N)}_{α_1,α_2,...,α_N}(λ)Θα1​,α2​,...,αN​(N)​(λ) is recurrently defined by closed formula.
A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.
A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.
Abstract In 1992, Scholtz et al. ( Ann. Phys., 213 (1992) 74) showed that a set of non-Hermitian operators can represent observables of a closed unitary quantum system, provided only that its elements are quasi-Hermitian ( i.e. , roughly speaking, Hermitian with respect to an ad hoc inner-product metric). We show that such a version of quantum mechanics admits a simultaneous closed-form representation of the metric and of the observables , in terms of auxiliary operators Z k with . At N = 2 the formalism degenerates to the well-known quantum mechanics using factorized metric , where is parity and where is charge.
During recent developments in quantum theory it has been clarified that observable quantities (such as energy and position) may be represented by operators $\mathrm{\ensuremath{\Lambda}}$ (with real spectra) that are manifestly non-Hermitian in a preselected friendly Hilbert space H(F){\mathcal{H}}^{(F)}H(F). The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition ${\mathcal{H}}^{(F)}\ensuremath{\rightarrow}{\mathcal{H}}^{(S)}$ to another, standard, Hilbert space. We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators ${\mathrm{\ensuremath{\Lambda}}}_{0}$ and ${\mathrm{\ensuremath{\Lambda}}}_{1}$) in advance, such an upgrade need not exist in general.