Table of contents

This special issue contains contributions presented at the workshop on The Physics of Non-Hermitian Operators held at Stellenbosch, South Africa in November 2005. A few additional articles are included from researchers who could not attend the meeting.

The workshop was the fourth meeting in a series on Pseudo-Hermitian Hamiltonians in Quantum Physics which commenced in Prague in 2003 as a result of growing interest in non-Hermitian Hamiltonians in general, and PT-symmetric quantum mechanics and quantum field theory in particular. Naturally, Hamiltonians of open systems are an obvious part of this topic, but the focus here is rather on systems where unitary time evolution as such is not undermined. This rather new, and perhaps ambitious, field thus explores a possible generalization of quantum mechanics which incorporates non-Hermitian Hamiltonians into a consistent quantum mechanical framework.

New results of a physical and mathematical nature have brought about appreciable progress; earlier work has been uncovered or even rediscovered and links between sub-fields have been established. This issue should provide a stimulus for initiating further advances in the field, re-focus on earlier perspectives and explore open and controversial questions. While non-Hermitian operators have been encountered in theoretical physics since the advent of quantum mechanics, their systematic mathematical investigation and their possible physical implications is a relatively new subject, further explored in this issue.

We would like to thank the University of Stellenbosch and the South African National Research Foundation for their financial support, without which it would not have been possible to plan and organize the workshop. Organizational aspects and technical support were very ably taken care of by Christine Ruperti and Tinus Botha whom we would like to thank again. Finally, we also appreciate and acknowledge the invitation from the Editorial Board of Journal of Physics A: Mathematical and General for the opportunity to publish this special issue and trust that this will indeed widen the interest and stimulate progress in the field.

Investigations during the last few years show that complex PT-symmetric or pseudo-Hermitian Hamiltonians possess a real discrete spectrum and several other features akin to a Hermitian Hamiltonian. These developments not only show us new directions but also demand a revisit to the conventional physics wherein PT-symmetry could have been invoked. After a review of some very interesting results, we present the super-barrier reflectionlessness and the cranking model of the nucleus. These two instances are being viewed as the meeting ground of the two: PT-symmetry and Hermiticity. The former presents an agreement and the latter is presently a dichotomy.

Motivated by the generalization of quantum theory for the case of non-Hermitian Hamiltonians with PT symmetry, we show how a classical cosmological model describes a smooth transition from ordinary dark energy to the phantom one. The model is based on a classical complex Lagrangian of a scalar field. Specific symmetry properties analogous to PT in non-Hermitian quantum mechanics lead to purely real equations of motion.

No-core shell model (NCSM) calculations using ab initio effective interactions are very successful in reproducing experimental nuclear spectra. The main theoretical approach is the use of effective operators, which include correlations left out by the truncation of the model space to a numerically tractable size. We review recent applications of the effective operator approach, within a NCSM framework, to the renormalization of the nucleon–nucleon interaction, as well as scalar and tensor operators.

This paper is a mini-summary of four new advances in the theory of -symmetric quantum mechanics. It describes some new calculations that were completed in 2005. The first advance concerns the classical coordinate-space trajectories in some -symmetric theories. Depending on the initial conditions, one can find arbitrarily long periodic -symmetric classical trajectories. The longer trajectories originate from smaller regions of initial conditions. There is an interesting resemblance to the so-called period-doubling route to chaos. The second advance concerns the perturbative construction of the operator for a -symmetric square well. The result has notable similarities to and differences from that for the -symmetric cubic oscillator. The third advance is a detailed comparison of a -symmetric Hamiltonian and the corresponding Hermitian Hamiltonian. It is argued that a perturbative construction of the Hermitian Hamiltonian leads to an almost intractable theory. The fourth advance concerns reflectionless potentials and symmetry and a possible connection with problems in cosmology.

For complex 2 × 2 matrices M, points of degeneracy (D), and chiral points (C) where the eigenvectors are circularly polarized, coincide when M is symmetric, and often lie close together when M is not symmetric. This proximity is explained statistically, by averaging over an ensemble in which the elements of M are random complex numbers: the standard deviation of the separation between C and D points is less than half the separation of the two D points. If the departure from symmetry is treated as a perturbation of strength ε, the separation between C and D points is of order ε²; the probability distribution of the separation is calculated.

In the framework of perturbation theory the reality of the perturbed eigenvalues of a class of symmetric Hamiltonians is proved using stability techniques. We apply this method to symmetric unperturbed Hamiltonians perturbed by symmetric additional interactions.

I review aspects of work done in collaboration with A Zee and R Scalettar (1997 Nucl. Phys. B 504 579; 1997 Nucl. Phys. B 501 643; 2001 J. Math. Phys.42 5718) on complex non-Hermitian random matrices. I open by explaining why the bag of tools used regularly in analysing Hermitian random matrices cannot be applied directly to analyse non-Hermitian matrices, and then introduce the method of Hermitization, which solves this problem. Then, for rotationally invariant ensembles, I derive a master equation for the average density of eigenvalues in the complex plane, in the limit of infinitely large matrices. This is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the Hermitized Green function. Remarkably, this resummation can be carried out explicitly for any rotationally invariant ensemble. I prove that in the limit of infinitely large matrices, the shape of the eigenvalue distribution is either a disc or an annulus. This is the celebrated 'single-ring' theorem. Which of these shapes is realized is determined by the parameters (coupling constants) which determine the ensemble. By varying these parameters a phase transition may occur between the two possible shapes. I briefly discuss the universal features of this transition. As the analysis of this problem relies heavily on summation of planar Feynman diagrams, I make a special effort at presenting a pedagogical exposition of the diagrammatic method, which some readers may find useful.

The spectrum of the spherically symmetric α²-dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant α-profiles, a perturbation theory and a Galerkin technique are developed in a Krein space approach. With the help of these tools, a very pronounced α-resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh. Non-oscillatory as well as oscillatory dynamo regimes are obtained. An estimation technique is developed for obtaining the critical α-profiles at which the eigenvalues enter the right spectral half-plane with non-vanishing imaginary components (at which overcritical oscillatory dynamo regimes form). Finally, Fréchet derivative (gradient) based methods are developed, suitable for further numerical investigations of Krein space related setups such as MHD α²-dynamos or models of -symmetric quantum mechanics.

Exceptional points for a two-dimensional matrix Hamiltonian that breaks time reversal symmetry are revisited. Particular attention is paid to C-points where the wavefunction exhibits circular polarization. They are different from exceptional points if and when time reversal symmetry is broken, or in crystal optics, when the crystal is optically active.

The thermodynamic limit of the Lipkin model is investigated. While the limit turns out to be rather elusive, the analysis gives strong indications that the limit yields two analytically dissociated operators, one for the normal and one for the deformed phase. While the Lipkin Hamiltonian is Hermitian and has a second-order phase transition in finite dimensions (finite particle number), both properties seem to be destroyed in the thermodynamic limit.

We numerically solved the implicit, transcendental equation that defines the eigenenergy surface of a degenerating isolated doublet of unbound states in the simple but illustrative case of the scattering of a beam of particles by a double barrier potential. Unfolding the degeneracy point with the help of a contact equivalent approximant, crossings and anticrossings of energies and widths, as well as the changes of identity of the poles of the S-matrix, are explained in terms of sections of the eigenenergy surfaces.

We present some unexpected links between the spectral properties of ensembles of large Hermitian and non-Hermitian random matrices, within the formalism of free random variables. We address the task of adding non-Hermitian random matrices. To solve this problem, we present a new approach [1] based on a generalization of the notion of the matrix-valued Green's function to the quaternion level, and on an extension of free probability theory, which allows us to introduce the setting of quaternion-valued free probability theory. We use this quaternion construction to solve the problem of a non-Hermitian random matrix of the type H₁ + iH₂, with Hermitian H₁, H₂ freely independent. Finally, we mention conformal mapping relating spectral properties of Hermitian H₁ + H₂ to spectral properties of the non-Hermitian H₁ + iH₂ model.

In the context of non-Hermitian but PT-symmetric Hamiltonians, we explore the systematics of the coefficients of the multiple commutators that appear in the perturbative expansions for Q, the exponent of the metric operator, and h, the equivalent Hermitian Hamiltonian. We find exact expressions for both Q and h for PT-symmetric versions of the dual two-dimensional quantum field theories, the sine-Gordon model and the massive Thirring model, and elucidate how these solutions arise from the structure of the multiple commutator coefficients.

We enlarge the limited class of analytically solvable one-dimensional periodic potentials with a finite number of band gaps by considering the complex, PT-symmetric generalized associated Lamé (GAL) potential with four parameters and applying supersymmetry. We obtain three duality relations which relate the band-edge eigenstates of the twenty four GAL potentials obtained by permutation of the four parameters.

We introduce a very simple, exactly solvable -symmetric non-Hermitian model with a real spectrum, and derive a closed formula for the metric operator which relates the problem to a Hermitian one.

We study the average density of resonances, ⟨ρ(x, y)⟩, in a semi-infinite disordered chain coupled to a perfect lead. The function ⟨ρ(x, y)⟩ is defined in the complex energy plane and the distance y from the real axes determines the resonance width. We concentrate on strong disorder and derive the asymptotic behaviour of ⟨ρ(x, y)⟩ in the limit of small y.

The physically admissible solutions of the -symmetric Scarf I potential are identified in the domain of real and complex energies. It is found that generally there are no admissible complex-energy solutions, and there is one with real energy. In a limited range of the parameters there are two series of seemingly admissible solutions both with the real and complex energies belonging to quasi-parity q = ±; however, the two sets are not -orthogonal in the domain of real energies. The sign of the pseudo-norm of states with real energy is found to oscillate as (−1)ⁿ, in accordance with the example of other -symmetric potentials possessing an infinite number of discrete levels. It is argued that the spontaneous breakdown of -symmetry cannot be defined for the Scarf I potential. A comparison with some -symmetric extensions of the infinite square well is presented.

We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, , in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1 + 1)-dimensions. For the case that v(x) is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v(x) = ix³. This reveals an infinite class of previously unknown - as well as non--inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in that is consistent with the usual choice for the quantum Hamiltonian. Finally, we discuss subtleties involved with higher dimensions and systems having a fixed length scale.

The rationale for introducing non-Hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-Hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-Hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to non-perturbative properties and the metric.

We consider quantum mechanics with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells, in particular, biorthogonal bases. The 'self-orthogonality' phenomenon is clarified in terms of a correct spectral decomposition and it is shown that 'self-orthogonal' states never jeopardize a resolution of identity and thereby quantum averages of observables. The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated. Some puzzles with zero-binorm bound states in a continuous spectrum are unravelled with the help of a correct resolution of identity.

Schrödinger's equation is solved numerically using the spectral collocation (pseudospectral) method based on Hermite weighted polynomial interpolants. Several sets of numerical results from the literature are reproduced using this approach. MATLAB codes that can serve as templates for further exploration are provided both in the paper and online. A new proposal is made for solving Schrödinger's equation in Stokes wedges.

A fundamental problem in the theory of -invariant quantum systems is to determine whether a given system 'respects' this symmetry or not. If not, the system usually develops non-real eigenvalues. It is shown in this contribution how to algorithmically detect the existence of complex eigenvalues for a given PT-symmetric matrix. The procedure uses classical results from stability theory which qualitatively locate the zeros of real polynomials in the complex plane. The interest and value of the present approach lies in the fact that it avoids diagonalization of the Hamiltonian at hand.

Discrete -symmetric square wells are studied. Their wavefunctions are found to be proportional to classical Tshebyshev polynomials of complex argument. The compact secular equations for energies are derived giving the real spectra in certain intervals of non-Hermiticity strengths Z. It is amusing to note that although the known square well re-emerges in the usual continuum limit, a twice as rich, upside-down symmetric spectrum is exhibited by all its present discretized predecessors.