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arXiv:0908.4045 · Symmetry Integrability and Geometry Methods and Applications

Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators

Miloslav Znojil

Published Aug 27, 2009• 7 citations
arXivHTMLPDF

Abstract

One-dimensional unitary scattering controlled by non-Hermitian (typically, PT -symmetric) quantum Hamiltonians H = H † is considered.Treating these operators via Runge-Kutta approximation, our three-Hilbert-space formulation of quantum theory is reviewed as explaining the unitarity of scattering.Our recent paper on bound states [Znojil M., SIGMA 5 (2009), 001, 19 pages, arXiv:0901.0700] is complemented by the text on scattering.An elementary example illustrates the feasibility of the resulting innovative theoretical recipe.A new family of the so called quasilocal inner products in Hilbert space is found to exist.Constructively, these products are all described in terms of certain nonequivalent short-range metric operators Θ = I represented, in Runge-Kutta approximation, by (2R -1)-diagonal matrices.

DOI
10.3842/sigma.2009.085

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One-dimensional unitary scattering controlled by non-Hermitian (typically, PT -symmetric) quantum Hamiltonians H = H † is considered.Treating these operators via Runge-Kutta approximation, our three-Hilbert-space formulation of quantum theory is reviewed as explaining the unitarity of scattering.Our recent paper on bound states [Znojil M., SIGMA 5 (2009), 001, 19 pages, arXiv:0901.0700] is complemented by the text on scattering.An elementary example illustrates the feasibility of the resulting innovative theoretical recipe.A new family of the so called quasilocal inner products in Hilbert space is found to exist.Constructively, these products are all described in terms of certain nonequivalent short-range metric operators Θ = I represented, in Runge-Kutta approximation, by (2R -1)-diagonal matrices.
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.
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 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.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
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.
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.