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Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS₅ x S⁵ background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007.

In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited.

Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc.

We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London.

We hope this special issue will be useful to the newcomer as well as to the expert in the subject.

Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.

We investigate properties of the most general -symmetric non-Hermitian Hamiltonian of cubic order in the annihilation and creation operators as a ten-parameter family. For various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral Hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and Moyal products. We elaborate on the subtleties of this approach. For special choices of the ten parameters the Hamiltonian reduces to various models previously studied, such as to the complex cubic potential, the so-called Swanson Hamiltonian or the transformed version of the from below unbounded quartic −x⁴-potential. In addition, it also reduces to various models not considered in the present context, namely the single-site lattice Reggeon model and a transformed version of the massive sextic ±x⁶-potential, which plays an important role as a toy model to identify theories with vanishing cosmological constant.

Recently Bender, Brody, Jones and Meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small, when the time-evolution operator is taken to be non-Hermitian but -symmetric. Here we demonstrate that such phenomena can also be obtained for non-Hermitian Hamiltonians for which -symmetry is completely broken, i.e. dissipative systems. We observe that the effect of a tunable passage time can be achieved by projecting between orthogonal eigenstates by means of a time-evolution operator associated with a non-Hermitian Hamiltonian. It is not essential that this Hamiltonian is -symmetric.

Spectral properties of singular Sturm–Liouville operators of the form with the indefinite weight x ↦ sgn(x) on are studied. For a class of potentials with lim_|x|→∞V(x) = 0 the accumulation of complex and real eigenvalues of A to zero is investigated and explicit eigenvalue problems are solved numerically.

The Riemann equation u_t + uu_x = 0, which describes a one-dimensional accelerationless perfect fluid, possesses solutions that typically develop shocks in a finite time. This equation is symmetric. A one-parameter -invariant complex deformation of this equation, u_t − iu(iu_x) = 0 ( real), is solved exactly using the method of characteristic strips, and it is shown that for real initial conditions, shocks cannot develop unless is an odd integer. When is an odd integer, the shock-formation time is calculated explicitly.

A technique for constructing an infinite tower of pairs of symmetric Hamiltonians, and (n = 2, 3, 4, ...), that have exactly the same eigenvalues is described and illustrated by means of three examples (n = 2, 3, 4). The eigenvalue problem for the first Hamiltonian of the pair must be posed in the complex domain, so its eigenfunctions satisfy a complex differential equation and fulfill homogeneous boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions of the second Hamiltonian of the pair obey a real differential equation and satisfy boundary conditions on the real axis. This equivalence constitutes a proof that the eigenvalues of both Hamiltonians are real. Although the eigenvalue differential equation associated with is real, the Hamiltonian exhibits quantum anomalies (terms proportional to powers of ℏ). These anomalies are remnants of the complex nature of the equivalent Hamiltonian . For the cases n = 2, 3, 4 in the classical limit in which the anomaly terms in are discarded, the pair of Hamiltonians H_n,classical and K_n,classical have closed classical orbits whose periods are identical.

The coupling of non-Hermitian -symmetric Hamiltonians to standard Hermitian Hamiltonians, each of which individually has a real energy spectrum, is explored by means of a number of soluble models. It is found that in all cases the energy remains real for small values of the coupling constant, but becomes complex if the coupling becomes stronger than some critical value. For a quadratic non-Hermitian -symmetric Hamiltonian coupled to an arbitrary real Hermitian -symmetric Hamiltonian, the reality of the ground-state energy for small enough coupling constant is established up to second order in perturbation theory.

In diffraction of a plane wave by a non-Hermitian PT symmetric optical lattice, the sum of the Bragg beam intensities need not be conserved, even though the gain and loss are equally distributed: the evolution is not unitary. Instead, different sums are conserved, in which the intensities are weighted with real numbers (positive or negative); several such sum rules are derived. Two-beam diffraction from a refractive index of the form constant −a cos x + ib sin x is studied in detail; the sum rule depends on the balance between the (real) Hermitian parameter a and the (real) anti-Hermitian parameter b.

We study nonlinear perturbative expansions for -symmetric local Schrödinger operators. The Schrödinger operator is a sum of the harmonic oscillator Hamiltonian and a local -symmetric potential depending, in general, nonlinearly on the perturbation parameter. A specific class of models having real spectrum for any value of the parameter is proposed.

The nonlinear flow equations discussed recently by Bender and Feinberg are all reduced to the well-known Euler equation after a change of variables.

The analytic structure in the vicinity of three coalescing eigenvalues (EP3) of a matrix problem is investigated. It is argued that the three eigenfunctions—also coalescing at the EP3—invoke a true chiral behaviour in the vicinity of the EP3 and that they can be related to a three-dimensional helix. The orientation of the helix depends on the distribution of the widths of the three levels in the vicinity of the EP3.

An algebraic technique useful in studying non-Hermitian Hamiltonians with real spectra, is presented. The method is illustrated by explicit application to a family of one-dimensional potentials.

In a previous paper [1] we introduced a very simple -symmetric non-Hermitian Hamiltonian with a real spectrum and derived a closed formula for the metric operator relating the problem to a Hermitian one. In this paper we propose an alternative formula for the metric operator, which we believe is more elegant and whose construction—based on a backward use of the spectral theorem for self-adjoint operators—provides new insights into the nature of the model.

We present a numerical study of the spectrum of the Laplacian in an unbounded strip with -symmetric boundary conditions. We focus on non-Hermitian features of the model reflected in an unusual dependence of the eigenvalues below the continuous spectrum on various boundary-coupling parameters.

Starting on the basis of the non-commutative q-differential calculus, we introduce a generalized q-deformed Schrödinger equation. It can be viewed as the quantum stochastic counterpart of a generalized classical kinetic equation, which reproduces at the equilibrium the well-known q-deformed exponential stationary distribution. In this framework, q-deformed adjoint of an operator and q-Hermitian operator properties occur in a natural way in order to satisfy the basic quantum mechanics assumptions.

The mechanism of introducing non-Hermiticity to non-central -symmetric potentials through both the φ azimuth and θ polar angles is discussed. Generalizing the results of a previous work it is shown that this can be done also through the polar angle part if appropriate potentials, such as the Scarf I or Rosen–Morse I potentials are used in the eigenvalue equation of the polar component. It is shown that the spontaneous breakdown of symmetry can also occur in these non-central potentials. Several potentials are proposed in the azimuthal eigenvalue equation too, where the use of periodic boundary conditions is essential. Possible generalizations of the results are outlined.

We analyze the zero energy solutions, of a two-dimensional system which undergoes a non-radial symmetric, complex potential V(r, ϕ). By virtue of the coherent states concept, the localized states are constructed, and the consequences of the imaginary part of the potential are found both analytically and schematically.

We give a simple proof of the fact that every diagonalizable operator that has a real spectrum is quasi-Hermitian and show how the metric operators associated with a quasi-Hermitian Hamiltonian are related to the symmetry generators of an equivalent Hermitian Hamiltonian.

Open quantum systems are embedded in the continuum of scattering wavefunctions and are naturally described by non-Hermitian Hamilton operators. In the complex energy plane, exceptional points appear at which two (or more) eigenvalues of the Hamilton operator coalesce. Although they are a countable set of single points in the complex energy plane and therefore of measure zero, they determine decisively the dynamics of open quantum systems. A powerful method for the description of open quantum systems is the Feshbach projection operator formalism. It is used in the present paper as a basic tool for the study of exceptional points and of the role they play for the dynamics of open quantum systems. Among others, the topological structure of the exceptional points, the rigidity of the phases of the eigenfunctions in their vicinity, the enhancement of observable values due to the reduced phase rigidity and the appearance of phase transitions are considered. The results are compared with existing experimental data on microwave cavities. In the last section, some questions being still unsolved, are considered.

We present closed form solutions to a certain class of one- and two-dimensional nonlinear Schrödinger equations involving potentials with broken and unbroken symmetry. In the one-dimensional case, these solutions are given in terms of Jacobi elliptic functions, hyperbolic and trigonometric functions. Some of these solutions are possible even when the corresponding -symmetric potentials have a zero threshold. In two-dimensions, hyperbolic secant type solutions are obtained for a nonlinear Schrödinger equation with a non-separable complex potential.

A complexified von Roos Hamiltonian is considered and a Hermitian first-order intertwining differential operator is used to obtain the related position-dependent mass η-weak-pseudo-Hermitian Hamiltonians. Using a Liouvillean-type change of variables, the η-weak-pseudo-Hermitian von Roos Hamiltonians H_x are mapped into the traditional Schrödinger Hamiltonian form H_q, where exact isospectral correspondence between H_x and H_q is obtained. Under a 'user-friendly' position-dependent-mass setting, it is observed that for each exactly solvable η-weak-pseudo-Hermitian reference-Hamiltonian H_q there is a set of exactly solvable η-weak-pseudo-Hermitian isospectral target-Hamiltonians H_x. A non-Hermitian -symmetric Scarf II and a non-Hermitian periodic-type -symmetric Samsonov–Roy potentials are used as reference models and the corresponding η-weak-pseudo-Hermitian isospectral target-Hamiltonians are obtained.

Recently developed methods for PT-symmetric models can be applied to quantum-mechanical matrix and vector models. In matrix models, the calculation of all singlet wavefunctions can be reduced to the solution of a one-dimensional PT-symmetric model. The large-N limit of a wide class of matrix models exists, and properties of the lowest-lying singlet state can be computed using WKB. For models with cubic and quartic interactions, the ground-state energy appears to show rapid convergence to the large-N limit. For the special case of a quartic model, we find explicitly an isospectral Hermitian matrix model. The Hermitian form for a vector model with O(N) symmetry can also be found, and shows many unusual features. The effective potential obtained in the large-N limit of the Hermitian form is shown to be identical to the form obtained from the original PT-symmetric model using familiar constraint field methods. The analogous constraint field prescription in four dimensions suggests that PT-symmetric scalar field theories are asymptotically free.

A harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part and its su(1,1) generalizations, for which a family of positive-definite metric operators was recently constructed, are re-examined in a supersymmetric context. Some quasi-Hermitian supersymmetric extensions of such Hamiltonians are proposed by enlarging su(1,1) to a superalgebra. This allows the construction of new non-Hermitian Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.

For a flux qubit considered as a two-level system, for which a hidden polynomial pseudo-supersymmetry was previously discovered, we propose a special time-dependent external control field. We show that for a qubit placed in this field there exists a critical value of tunnel frequency. When the tunnel frequency is close enough to its critical value, the external field frequency may be tuned in a way to keep the probability to detect a definite direction of the current circulating in a Josephson-junction circuit above 1/2 during a desirable time interval. We also show that such a behavior is not much affected by a sufficiently small dissipation.

We show that the complex projections of time-dependent η-quasianti-Hermitian quaternionic Hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi-Hermitian density matrices if and only if a quasistationarity condition is fulfilled, i.e. if and only if η is an Hermitian positive time-independent complex operator. An example is also discussed.

We explore some aspects of -symmetric Hamiltonians with two point interactions. We determine classes of point interactions for which the Hamiltonians are supersymmetric. We prove that these Hamiltonians are quasi-Hermitian and find a very simple formula for the metric operator Θ and its square root ϱ as well. Further, we present the quasi-Hermitian Hamiltonian (with one-point interaction) with a continuous spectrum.

We discuss the Hamiltonian H = p²/2 − (ix)²ⁿ⁺¹ and the mixed Hamiltonian H_mixed = (p² + x²)/2 − g(ix)²ⁿ⁺¹. The Hamiltonians H and in some cases also H_mixed are crypto-Hermitian in a sense that, in spite of their apparent non-Hermiticity, a quantum spectral problem can be formulated such that the spectrum is real. We note that the corresponding classical Hamiltonian system can be treated as a gauge system, with the imaginary part of the Hamiltonian playing the role of the first class constraint. Several different nontrivial quantum problems can be formulated on the basis of this classical problem. We formulate and solve some such problems. We consider then the mixed Hamiltonian and find that its spectrum undergoes in certain cases a rather amazing transformation when the coupling g is sent to zero. There is an infinite set of exceptional points g^(j)_⋆ where a couple of eigenstates of H coalesce and their eigenvalues cease to be real. When quantization is done in the most natural way such that gauge constraints are imposed on quantum states, the spectrum should not be positive definite, but must involve the negative energy states (ghosts). We speculate that, in spite of the appearance of ghost states, unitarity might still be preserved.

Although the quantum bound-state energies may be generated by the so-called -symmetric Hamiltonians where is, typically, parity, the spectrum only remains real and observable (i.e., in the language of physics, the -symmetry remains unbroken) inside a domain of couplings. We show that the boundary (i.e., certain stability and observability horizon formed by Kato's exceptional points) remains algebraic (i.e., we determine it by closed formulae) for a certain toy-model family of N-dimensional anharmonic-oscillator-related matrix Hamiltonians H^(N) with N = 2, 3, ..., 11.