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arXiv:0903.1007 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Scattering theory using smeared non-Hermitian potentials

Miloslav Znojil

Published Aug 12, 2009• 41 citations
arXivHTMLPDF

Abstract

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 $x$ [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.

DOI
10.1103/physrevd.80.045009

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Local non-Hermitian potentials V(x)\ensuremath≠V∗(x)V(x)\ensuremath{\ne}{V}^{*}(x)V(x)\ensuremath=V∗(x) can, sometimes, generate stable bound states \ensuremathψ(x)\ensuremath{\psi}(x)\ensuremathψ(x) at real energies. Unfortunately, the idea [based on the use of a non-Dirac ad hoc metric \ensuremathΘ(x,x\ensuremath′)\ensuremath≠\ensuremathδ(x\ensuremath−x\ensuremath′)\ensuremath{\Theta}(x,{x}^{\ensuremath{'}})\ensuremath{\ne}\ensuremath{\delta}(x\ensuremath{-}{x}^{\ensuremath{'}})\ensuremathΘ(x,x\ensuremath′)\ensuremath=\ensuremathδ(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≠V\ifmmode†\else\textdagger\fiV\ensuremath{\ne}{V}^{\ifmmode\dagger\else\textdagger\fi{}}V\ensuremath=V\ifmmode†\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.
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.
We highlight the conceptual issues that arise when one applies the quasi-Hermitian framework to analyze scattering from localized non-Hermitian potentials, in particular complex square wells or delta functions. When treated in the framework of conventional quantum mechanics, these potentials are generally considered as effective theories, in which probability is not conserved because of processes that have been ignored. However, if they are treated as fundamental theories, the Hilbert-space metric must be changed. In order for the newly defined probability to be conserved, it must differ from the standard one, even at asymptotically large distances from the scattering center, and the mechanism for this is the nonlocality of the new metric, as we show in detail in the model of a single complex delta function. However, properties of distant bound-state systems, which do not interact physically with the non-Hermitian scattering potential, should not be affected. We analyze a model Hamiltonian that supports this contention.
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.
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, h→H=Ω h Ω≠H†\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\daggerh→H=ΩhΩ=H† induces the change L→K{\cal L} \to {\cal K}L→K of the Hilbert space of states, reflected by the loss of the Hermiticity of H≠H†H\neq H^\daggerH=H†. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts K{\cal K}K into the third, correct physical Hilbert space H{\cal H}H, unitarily equivalent to L{\cal L}L. The situation encountered, typically, in PT−{\cal PT}-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.
Abstract A family of exactly solvable quantum square wells with discrete coordinates and with certain non-stationary Hermiticity-violating Robin boundary conditions is proposed and studied. Manifest non-Hermiticity of the model in conventional Hilbert space H friendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space H physical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ(t)) as well as the time-dependent observables (say, Λ(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrödinger-equation generator) and Σ(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in H physical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + Σ(t). The applicability of these ideas is illustrated via an N by N matrix model. At N = 2, closed formulae are presented not only for the measurable instantaneous energy spectrum but also for all of the eligible time-dependent physical inner-product metrics Θ(N=2)(t), for the related Dyson maps Ω(N=2)(t), for the Coriolis force Σ(N=2)(t) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrödinger-equation generator G (N=2)(t).
We consider the interaction between the Hermitian world, represented by a real delta-function potential $\ensuremath{-}\ensuremath{\alpha}\ensuremath{\delta}(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $i\ensuremath{\beta}(\ensuremath{\delta}(x\ensuremath{-}L)\ensuremath{-}\ensuremath{\delta}(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave function is small for LLL large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric $\ensuremath{\eta}$, and consequently a new wave function $\ensuremath{\Psi}(x)$, defined in terms of the original wave function $\ensuremath{\psi}(x)$ by means of $\ensuremath{\eta}$. In this picture, working perturbatively in $\ensuremath{\beta}$, the bound-state wave function is actually unchanged from its unperturbed form for $|x|\ensuremath{\ll}L$. However, the scattering wave function, for $|x|\ensuremath{\gg}L$, is changed in a significant manner. In particular, there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.