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Hermitian/non-Hermitian interfaces in Schroedinger picture

Miloslav Znojil

Published Oct 10, 2017• 0 citations
OpenAlex

Abstract

In any unitary quantum theory with Hilbert space metric $\Theta$ the Schr\{o}dinger equation may be either Hermitian ($\Theta=I$) or non-Hermitian ($\Theta \neq I$). For local potentials the corresponding classes of quantum systems are clearly separated and the two families of the respective Hamiltonians have an empty overlap. Via elementary solvable examples we demonstrate that whenever the potentials are chosen weakly nonlocal, the separation may disappear and the overlaps {\it alias} interfaces may become non-empty.

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In any unitary quantum theory with Hilbert space metric Θ\ThetaΘ the Schr{o}dinger equation may be either Hermitian (Θ=I\Theta=IΘ=I) or non-Hermitian (Θ≠I\Theta \neq IΘ=I). For local potentials the corresponding classes of quantum systems are clearly separated and the two families of the respective Hamiltonians have an empty overlap. Via elementary solvable examples we demonstrate that whenever the potentials are chosen weakly nonlocal, the separation may disappear and the overlaps {\it alias} interfaces may become non-empty.
In the global framework of quantum theory, the individual quantum systems seem clearly separated into two families with the respective manifestly Hermitian and hiddenly Hermitian operators of their Hamiltonian. In the light of certain preliminary studies, these two families seem to have an empty overlap. In this paper, we will show that whenever the interaction potentials are chosen to be weakly nonlocal, the separation of the two families may disappear. The overlapsaliasinterfaces between the Hermitian and non-Hermitian descriptions of a unitarily evolving quantum system in question may become nonempty. This assertion will be illustrated via a few analytically solvable elementary models.
In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψψψ is controlled, in Hilbert space L{\cal L}L, by a Hamiltonian h\mathfrak{h}h which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces h\mathfrak{h}h by its isospectral but non-Hermitian avatar H=Ω−1hΩH = Ω^{-1}\mathfrak{h}ΩH=Ω−1hΩ with Ω†Ω=Θ≠IΩ^\daggerΩ= Θ\neq IΩ†Ω=Θ=I. Although acting in another, manifestly unphysical Hilbert space H{\cal H}H, the amended Hamiltonian H≠H†H \neq H^\daggerH=H† can be perceived as self-adjoint with respect to the amended inner-product metric ΘΘΘ. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of HHH we introduce and describe a specific new family of Hamiltonians HHH for which the metrics ΘΘΘ 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 L{\mathcal{L}}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 H{\mathcal{H}}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 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.
Local non-Hermitian potentials $V(x)\ensuremath{\ne}{V}^{*}(x)$ can, sometimes, generate stable bound states $\ensuremath{\psi}(x)$ at real energies. Unfortunately, the idea [based on the use of a non-Dirac ad hoc metric $\ensuremath{\Theta}(x,{x}^{\ensuremath{'}})\ensuremath{\ne}\ensuremath{\delta}(x\ensuremath{-}{x}^{\ensuremath{'}})$ in Hilbert space] cannot directly be transferred to scattering due to the related loss of the asymptotic observability of xxx [cf. H. F. Jones, Phys. Rev. D 78, 065032 (2008)]. We argue that for smeared (typically, nonlocal or momentum-dependent) potentials $V\ensuremath{\ne}{V}^{\ifmmode\dagger\else\textdagger\fi{}}$ this difficulty may be circumvented. A return to the usual (i.e., causal and unitary) quantum scattering scenario is then illustrated via an exactly solvable multiple-scattering example. In it, the anomalous loss of observability of the coordinate remains restricted to a small vicinity of the scattering centers.
Excited bound states are often understood within scattering-based theories as resulting from the collision of a particle on a target via a short-range potential. We show that the resulting formalism is non-Hermitian and describe the Hilbert spaces and metric operator relevant to a correct formulation of such theories. The structure and tools employed are the same that have been introduced in current works dealing with PT-symmetric and quasi-Hermitian problems. The relevance of the non-Hermitian formulation to practical computations is assessed by introducing a non-Hermiticity index. We give a numerical example involving scattering by a short-range potential in a Coulomb field for which it is seen that even for a small but non-negligible non-Hermiticity index the non-Hermitian character of the problem must be taken into account. The computation of physical quantities in the relevant Hilbert spaces is also discussed.