N-site-lattice analogues of V(x)=ix3

doi:10.1016/j.aop.2011.12.009

Annals of Physics

Volume 327, Issue 3, March 2012, Pages 893-913

https://doi.org/10.1016/j.aop.2011.12.009 Get rights and content

Abstract

A discrete $N$ -level alternative to the popular imaginary cubic oscillator is proposed and studied. As usual, the unitarity of evolution is guaranteed by the introduction of an ad hoc, Hamiltonian-dependent inner-product metric, the use of which defines the physical Hilbert space and renders the Hamiltonian (with real spectrum) observable. Due to the simplicity of our model of dynamics the construction of the set of eligible metrics is shown tractable by non-numerical means which combine the computer-assisted algebra with the extrapolation and/or perturbation techniques.

Highlights

► Elementary non-Hermitian quantum-lattice Hamiltonians $H (a, z)$ with real spectra proposed. ► Exceptional points $a = α$ calculated. ► All topologically nonequivalent stability-loss patterns shown reachable via “exponent” $z$ . ► “Critical exponents” of PT-symmetry breakdowns assigned Fibonacci-related classification and calculated. ► Hermitization matrices $Θ$ shown obtainable (and samples constructed) via symbolic manipulations and extrapolations in $N$ .

Introduction

Many measured spectra of energies may be interpreted as excitations of a quasiparticle. The simplest fits of such a type employ the elementary one-dimensional differential Schrödinger equation $- \frac{d^{2}}{d x^{2}} ψ_{n} (x) + V (x) ψ (x) = E_{n} ψ_{n} (x) ψ (\pm Λ) = 0, Λ \leq \infty$ containing a real potential $V (x)$ . In more sophisticated models, potential $V (x)$ may even be allowed complex, provided only that the spectrum itself remains real (cf. reviews [1], [2], [3], [4] for details).

Whenever $V (x) \neq V^{*} (x)$ , the reality of the spectrum may be fragile and sensitive to perturbations [5]. A remarkable exception emerges with the robustly real spectra generated by many potentials with the property $V (x) = V^{*} (- x)$ called, in the literature, $PT$ -symmetry [2] alias parity–pseudo-Hermiticity [3] alias Krein-space-Hermiticity [6].

In the early studies of this remarkable mathematical phenomenon the attention of the authors has mainly been restricted to the imaginary cubic oscillator example $V (x) = V^{(I C)} (x) = i x^{3}$ modified, possibly, by some other, asymptotically subdominant terms (cf., e.g., papers by Caliceti et al. [7], Alvarez [8] or Bender et al. [9], [10] and several further authors [11]). In what follows, we shall mainly feel inspired by this choice of $V (x)$ as well.

The essence of our present message will lie in the recommendation of a drastic simplification of the necessary mathematics. In a way explained in Section 2 this will be achieved by means of the replacement of the differential Schrödinger equation (1) by its discrete, difference-equation analogue. A few simplest illustrations will be then added in Section 3 where the discretization of the coordinate will be shown to facilitate the study of the reality (i.e., in principle, observability) of the spectrum.

The desirable flexibility of our present discrete simulation of the dynamical energy spectra will be shown achieved by a supplementary one-parametric deformation of the potential resembling slightly the influential proposal of deformation $i x^{3} \to {(i x)}^{3 + δ}$ by Bender and Boettcher [10]. The details will be described in Section 4. The subsequent Section 5 will then recall the known theory and explain some of its details via the simplest possible example with $N = 2$ . In Sections 6 The first nontrivial, 7 The, 8 Extrapolations and models with we shall finally present some applications of this theory to the models with $N = 4, N = 6$ and general $N = 2 K \geq 8$ , respectively. Section 9 is summary.

Section snippets

Discrete Schrödinger equations

The spectrum of many Krein-space-Hermitian Hamiltonians $H \neq H^{†}$ has been found robustly real and bounded below [10], [12], [13]. In the spirit of the general theory as outlined, first, by Scholtz et al. [14], one can conclude that the apparent non-Hermiticity is “false” and that it may be reinterpreted as a mere consequence of an inappropriate choice of the Hilbert-space representation $L^{2} (R) \equiv H^{(F)}$ . Hence, the abstract remedy is straightforward and lies in an interpretation-mediating transition to a

The series of one-parametric Hamiltonians

Let us pick up the potential $V (x_{j}) = i x_{j}^{3}$ and insert it in Eq. (6). Once we abbreviate $a = h^{5} / 8$ we obtain the following one-parametric sequence of Hamiltonians $H^{(2)} (a) = (\begin{matrix} 2 - i a & - 1 \\ - 1 & 2 + i a \end{matrix}), H^{(3)} (a) = (\begin{matrix} 2 - 8 i a & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 2 + 8 i a \end{matrix}),$ $H^{(4)} (a) = (\begin{matrix} 2 - 27 i a & - 1 & 0 & 0 \\ - 1 & 2 - i a & - 1 & 0 \\ 0 & - 1 & 2 + i a & - 1 \\ 0 & 0 & - 1 & 2 + 27 i a \end{matrix}), \dots .$ The first element $H^{(2)} (a)$ of this series is particularly elementary. For this reason it has also been chosen and studied in the methodical part of Ref. [15] (cf. its section II. B).

For the sake of brevity we shall restrict our attention

The series of two-parametric Hamiltonians

In the examples of the preceding section we encounter another suspicious feature even at any fixed and finite dimension $N = 2 K$ . Indeed, the critical left and right points $a = \pm α^{(2 K)}$ of the loss of the reality of the spectrum seem to be exclusively connected with the confluence of the first and second lowest excited states $ε_{2}$ and $ε_{3}$ or, symmetrically, of their equally “privileged” mirror partners $ε_{N - 2}$ and $ε_{N - 1}$ .

In what follows we shall explain this apparent privilege as an artifact caused by the too

Interpretation

Usually [25] people decide to work in a fixed, specific Hilbert space $H^{(S)}$ and treat a given quantum system as physical if and only if its time evolution is unitary. Naturally, this is the strategy which does not change if the space $H^{(S)}$ proves endowed with a general, nontrivial metric $Θ = Θ^{†} > 0$ (i.e., with its inner product defined in terms of this metric, cf. Eq. (2) above). One still has to guarantee that every candidate for an operator of observable proves self-adjoint with respect to this

The complete set of pseudometrics

The general four-parametric Hermitian $N = 4$ candidates for the metric (= “pseudometrics”) may be obtained, most easily, from Eq. (10) again. These solutions appear symmetric with respect to their second diagonal. They may be written in the closed four-parametric form of matrix $Θ {[H^{(4)} (a, z)]}_{(k, m, r, h)}$ $= [\begin{matrix} k & m - i k w & W^{*} & Z^{*} \\ m + i k w & r & h - i (k w + r a) & W^{*} \\ W & h + i (k w + r a) & r & m - i k w \\ Z & W & m + i k w & k \end{matrix}]$ where we introduced function $w = w (z, a) = 3^{z} a$ (this quantity must be positive and, for $z > 0$ , larger than $a$ ) and where we abbreviated $W = W (k, m, r) = - w^{2} k + r -$

The $N = 6$ model

For the not too large matrix dimensions $N = 2 K$ the technique of the construction of the general pseudometrics $Θ$ via the solution of the linear algebraic equation Eq. (10) remains feasible and straightforward. With the growth of $N$ the only difficulty emerges in connection with the printed presentation of the resulting multiparametric set of pseudometrics. For this reason it makes sense to find a sufficiently representative set of a few key parameters. For each such choice, moreover, it becomes

Extrapolations and models with $N \geq 8$

At any dimension $N = 2 K$ the dynamical contents of our toy models is controlled by the Hamiltonian (which varies with the “kinematical” parameter $a \in (- α^{(N)}, α^{(N)})$ , “dynamical” parameter $z \in (- \infty, \infty)$ and real spectral locus ${ε_{j} (a, z)}$ ) and by the Hermitian, positive-definite metric $Θ^{(N)}$ specified by an $N$ -plet of parameters ${k, m, \dots} \in D$ .

Naturally, practical implementations of such a recipe will require also a determination of the whole positivity domain $D$ , or of its suitable non-empty subdomain at least. This

Summary

Our first tests of the idea of discretization proved disappointing. We observed that the growing-dimension series (7) of the simple-minded one-parametric descendants of the popular differential imaginary cubic oscillator do not offer a sufficiently rich variability of the coupling-dependence of the energy spectra. Fortunately, the merely slightly more sophisticated and re-scaled two-parametric choice (8) of the model has been shown to offer a flexibility of the spectral loci which covers a

Acknowledgments

This work is supported by the GAČR grant No. P203/11/1433, the MŠMT “Doppler Institute” project No. LC06002 and the Institutional Research Plan AV0Z10480505.

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Annals of Physics

N-site-lattice analogues of V(x)=ix3

Abstract

Highlights

Introduction

Section snippets

Discrete Schrödinger equations

The series of one-parametric Hamiltonians

The series of two-parametric Hamiltonians

Interpretation

The complete set of pseudometrics

The N=6 model

Extrapolations and models with N≥8

Summary

Acknowledgments

Czech. J. Phys.

SIGMA

J. Phys. A

Phys. Rev. D

Phys. Rev. D

J. Phys. A

Rep. Progr. Phys.

Int. J. Geom. Methods Mod. Phys.

SIGMA

J. Math. Phys.

Phys. Lett. B

Phys. Rev. A

J. Phys. A

Phys. Rev. A

Comm. Math. Phys.

J. Phys. A: Math. Gen.

Phys. Rev. D

Phys. Rev. Lett.

J. Phys. A: Math. Gen.

Phys. Lett. A

J. Phys. A

J. Phys. A: Math. Gen.

J. Phys. A: Math. Gen.

J. Phys. A: Math. Gen.

Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding

Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology

Wheeler-DeWitt Equation and the Applicability of Crypto-Hermitian Interaction Representation in Quantum Cosmology

Energy and momentum operator substitution method derived from Schrödinger equation for light and matter waves

Crypto-Hermitian approach to the Klein-Gordon equation

Crypto-hermitian approach to the klein–gordon equation

$N$ -site-lattice analogues of $V (x) = i x^{3}$

The $N = 6$ model

Extrapolations and models with $N \geq 8$