98 research outputs found
Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian
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 modelto identify theories
with vanishing cosmological constant.Comment: 21 page
Non-Hermitian Hamiltonians of Lie algebraic type
We analyse a class of non-Hermitian Hamiltonians, which can be expressed
bilinearly in terms of generators of a 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.Comment: 27 page
Metric operators for non-Hermitian quadratic su(2) Hamiltonians
A class of non-Hermitian quadratic su(2) Hamiltonians having an anti-linear
symmetry is constructed. This is achieved by analysing the possible symmetries
of such systems in terms of automorphisms of the algebra. In fact, different
realisations for this type of symmetry are obtained, including the natural
occurrence of charge conjugation together with parity and time reversal. Once
specified the underlying anti-linear symmetry of the Hamiltonian, the former,
if unbroken, leads to a purely real spectrum and the latter can be mapped to a
Hermitian counterpart by, amongst other possibilities, a similarity
transformation. Here, Lie-algebraic methods which were used to investigate the
generalised Swanson Hamiltonian are employed to identify the class of quadratic
Hamiltonians that allow for such a mapping to the Hermitian counterpart.
Whereas for the linear su(2) system every Hamiltonian of this type can be
mapped to a Hermitian counterpart by a transformation which is itself an
exponential of a linear combination of su(2) generators, the situation is more
complicated for quadratic Hamiltonians. Therefore, the possibility of more
elaborate similarity transformations, including quadratic exponents, is also
explored in detail. The existence of finite dimensional representations for the
su(2) Hamiltonian, as opposed to the su(1,1) studied before, allows for
comparison with explicit diagonalisation results for finite matrices. Finally,
the similarity transformations constructed are compared with the analogue of
Swanson's method for exact diagonalsation of the problem, establishing a simple
relation between both approaches.Comment: 25 pages, 6 figure
The quantum brachistochrone problem for non-Hermitian Hamiltonians
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 PT-symmetric. 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 with a non-Hermitian Hamiltonian. It is not essential that this Hamiltonian is PT-symmetric
Quantum isotonic nonlinear oscillator as a Hermitian counterpart of Swanson Hamiltonian and pseudo-supersymmetry
Within the ideas of pseudo-supersymmetry, we have studied a non-Hermitian
Hamiltonian H_{-}=\omega(\xi^{\dag} \xi+\1/2)+\alpha \xi^{2}+\beta \xi^{\dag
2}, where and is a first order differential
operator, to obtain the partner potentials and which are
new isotonic and isotonic nonlinear oscillators, respectively, as the Hermitian
equivalents of the non-Hermitian partner Hamiltonians . We have
provided an algebraic way to obtain the spectrum and wavefunctions of a
nonlinear isotonic oscillator. The solutions of which are Hermitian
counterparts of Swanson Hamiltonian are obtained under some parameter
restrictions that are found. Also, we have checked that if the intertwining
operator satisfies , where and is the first order differential operator,
which factorizes Hermitian equivalents of .Comment: 11 page
Integrable models from PT-symmetric deformations
We address the question of whether integrable models allow for PT-symmetric
deformations which preserve their intgrability. For this purpose we carry out
the Painleve test for PT-symmetric deformations of Burgers and the Korteweg-De
Vries equation. We find that the former equation allows for infinitely many
deformations which pass the Painleve test. For a specific deformation we prove
the convergence of the Painleve expansion and thus establish the Painleve
property for these models, which are therefore thought to be integrable. The
Korteweg-De Vries equation does not allow for deformations which pass the
Painleve test in complete generality, but we are able to construct a defective
Painleve expansion.Comment: 14 pages Late
Compactons versus Solitons
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 Painleve test fails, such that no soliton solutions can
be found. The Painleve 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.Comment: 4 page
A well-kept treasure at depth: precious red coral rediscovered in Atlantic deep coral gardens (SW Portugal) after 300 years
The highly valuable red coral Corallium rubrum is listed in several Mediterranean Conventions for species protection and management since the 1980s. Yet, the lack of data about its Atlantic distribution has hindered its protection there. This culminated in the recent discovery of poaching activities harvesting tens of kg of coral per day from deep rocky reefs off SW Portugal. Red coral was irregularly exploited in Portugal between the 1200s and 1700s, until the fishery collapsed. Its occurrence has not been reported for the last 300 years.info:eu-repo/semantics/publishedVersio
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