We analyse a class of non-Hermitian Hamiltonians, which can be expressed in terms of bilinear combinations of generators in the sl 2 (R)-Lie algebra or their isomorphic su(1, 1)-counterparts. The Hamlitonians are prototypes for solvable models of Lie algebraic type. Demanding a real spectrum and the existence of a well defined metric, we systematically investigate the constraints these requirements impose on the coupling constants of the model and the parameters in the metric operator. We compute isospectral Hermitian counterparts for some of the original non-Hermitian Hamiltonian. Alternatively we employ a generalized Bogoliubov transformation, which allows to compute explicitly real energy eigenvalue spectra for these type of Hamiltonians, together with their eigenstates. We compare the two approaches.Non-Hermitian Hamiltonians of Lie algebraic type sl 2 (C). It is well known that this algebra contains the compact real form su(2) and the non-compact real form sl 2 (R), which is isomorphic to su(1, 1), see for instance [33,34]. We will focus here on these two choices. Hamiltonians of sl 2 (R)-Lie algebraic typeThe three generator J 0 , J 1 and J 2 of sl 2 (R) satisfy the commutation relations [J 1 , J 2 ] = −iJ 0 , [J 0 , J 1 ] = iJ 2 and [J 0 , J 2 ] = −iJ 1 , such that the operators J 0 , J ± = J 1 ± J 2 obeyAs possible realisation for this algebra one may take for instance the differential operators2) allegedly attributed to Sophus Lie, see e.g. [31]. Clearly the action of this algebra on the space of polynomials V n = span{1, x, x 2 , x 3 , x 4 , ..., x n } (2.3) leaves it invariant. According to the above specified notions, a quasi-exactly solvable Hamiltonian of Lie algebraic type is therefore of the general form H J = l=0,± κ l J l + n,m=0,±

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 PTsymmetric. Here we demonstrate that such phenomena can also be obtained for non-Hermitian Hamiltonians for which PT -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 to a non-Hermitian Hamiltonian. It is not essential that this Hamiltonian is PT -symmetric.

We investigate properties of the most general PT -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 −x 4 -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 ±x 6 -potential, which plays an important role as a toy model to identify theories with vanishing cosmological constant.

We investigate whether the recently proposed PT -symmetric extensions of generalized Korteweg-de Vries equations admit genuine soliton solutions besides compacton solitary waves. For models which admit stable compactons having a width which is independent of their amplitude and those which possess unstable compacton solutions the Painlevé test fails, such that no soliton solutions can be found. The Painlevé test is passed for models allowing for compacton solutions whose width is determined by their amplitude. Consequently, these models admit soliton solutions in addition to compactons and are integrable.

This is the unspecified version of the paper.This version of the publication may differ from the final published version. Abstract: We provide a novel procedure to obtain complex PT -symmetric multi-particle Calogero systems. Instead of extending or deforming real Calogero systems, we explore here the possibilities for complex systems to arise from real nonlinear field equations. We exemplify this procedure for the Boussinesq equation and demonstrate how singularities in real valued wave solutions can be interpreted as N complex particles scattering amongst each other. We analyze this phenomenon in more detail for the two and three particle case. Particular attention is paid to the implemention of PT -symmetry for the complex multi-particle systems. New complex PT -symmetric Calogero systems together with their classical solutions are derived. Permanent