20 research outputs found
PT-symmetric deformations of Calogero models
We demonstrate that Coxeter groups allow for complex PT-symmetric deformations across the boundaries of all Weyl chambers. We compute the explicit deformations for the A2 and G2-Coxeter group and apply these constructions to Calogero–Moser–Sutherland models invariant under the extended Coxeter groups. The eigenspectra for the deformed models are real and contain the spectra of the undeformed case as subsystems
E2-quasi-exact solvability for non-Hermitian models
We propose the notion of E2-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the complex Mathieu Hamiltonian in a double scaling limit, which enables us to compute the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions are univariate polynomials in the energy obeying a three-term recurrence relation. The latter property guarantees the existence of a linear functional such that the polynomials become orthogonal. The polynomials are shown to factorize for all levels above the quantization condition leading to vanishing norms rendering them to be weakly orthogonal. In two concrete examples we compute the explicit expressions for the Stieltjes measure
Non-Hermitian multi-particle systems from complex root spaces
We provide a general construction procedure for antilinearly invariant
complex root spaces. The proposed method is generic and may be applied to any
Weyl group allowing to take any element of the group as a starting point for
the construction. Worked out examples for several specific Weyl groups are
presented, focusing especially on those cases for which no solutions were found
previously. When applied in the defining relations of models based on root
systems this usually leads to non-Hermitian models, which are nonetheless
physically viable in a self-consistent sense as they are antilinearly invariant
by construction. We discuss new types of Calogero models based on these complex
roots. In addition we propose an alternative construction leading to q-deformed
roots. We employ the latter type of roots to formulate a new version of affine
Toda field theories based on non-simply laced roots systems. These models
exhibit on the classical level a strong-weak duality in the coupling constant
equivalent to a Lie algebraic duality, which is known for the quantum version
of the undeformed case.Comment: 29 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
Antilinear deformations of Coxeter groups, an application to Calogero models
We construct complex root spaces remaining invariant under antilinear
involutions related to all Coxeter groups. We provide two alternative
constructions: One is based on deformations of factors of the Coxeter element
and the other based on the deformation of the longest element of the Coxeter
group. Motivated by the fact that non-Hermitian Hamiltonians admitting an
antilinear symmetry may be used to define consistent quantum mechanical systems
with real discrete energy spectra, we subsequently employ our constructions to
formulate deformations of Coxeter models remaining invariant under these
extended Coxeter groups. We provide explicit and generic solutions for the
Schroedinger equation of these models for the eigenenergies and corresponding
wavefunctions. A new feature of these novel models is that when compared with
the undeformed case their solutions are usually no longer singular for an
exchange of an amount of particles less than the dimension of the
representation space of the roots. The simultaneous scattering of all particles
in the model leads to anyonic exchange factors for processes which have no
analogue in the undeformed case.Comment: 32 page
Complex solitons with real energies
Using Hirota’s direct method and Bäcklund transformations we construct explicit complex one and two-soliton solutions to the complex Korteweg-de Vries equation, the complex modified Korteweg-de Vries equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be PT -symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex PT -symmetric solutions to the Korteweg-de Vries equation may also be constructed alternatively from real solutions to the modified Korteweg-de Vries by means of Miura transformations. The multi-soliton solutions obtained from Hirota’s method break the PT -symmetric, whereas those obtained from Bäcklund transformations are PT -invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are PT -symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space-time and the fact that any of our N-soliton solutions may be decomposed into N separate PT -symmetrizable one-soliton solutions
PT-symmetrically deformed shock waves
We investigate for a large class of nonlinear wave equations, which allow for
shock wave formations, how these solutions behave when they are
PT-symmetrically deformed. For real solutions we find that they are transformed
into peaked solutions with a discontinuity in the first derivative instead. The
systems we investigate include the PT-symmetrically deformed inviscid Burgers
equation recently studied by Bender and Feinberg, for which we show that it
does not develop any shocks, but peaks instead. In this case we exploit the
rare fact that the PT-deformation can be provided by an explicit map found by
Curtright and Fairlie together with the property that the undeformed equation
can be solved by the method of characteristics. We generalise the map and
observe this type of behaviour for all integer values of the deformation
parameter epsilon. The peaks are formed as a result of mapping the multi-valued
self-avoiding shock profile to a multi-valued self-crossing function by means
of the PT-deformation. For some deformation parameters we also investigate the
deformation of complex solutions and demonstrate that in this case the
deformation mechanism leads to discontinuties.Comment: 17 pages, 10 figure
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
Complex BPS solitons with real energies from duality
Following a generic approach that leads to Bogomolny–Prasad–Sommerfield (BPS) soliton solutions by imposing self-duality, we investigate three different types of non-Hermitian field theories. We consider a complex version of a logarithmic potential that possess BPS super-exponential kink and antikink solutions and two different types of complex generalizations of systems of coupled sine-Gordon models with kink and antikink solution of complex versions of arctan type. Despite the fact that all soliton solutions obtained in this manner are complex in the non-Hermitian theories we show that they possess real energies. For the complex extended sine-Gordon model we establish explicitly that the energies are the same as those in an equivalent pair of a non-Hermitian and Hermitian theory obtained from a pseudo-Hermitian approach by means of a Dyson map. We argue that the reality of the energy is due to the topological properties of the complex BPS solutions. These properties result in general from modified versions of antilinear CPT symmetries that relate self-dual and an anti-self-dual theories
PT-symmetric noncommutative spaces with minimal volume uncertainty relations
We provide a systematic procedure to relate a three dimensional q-deformed
oscillator algebra to the corresponding algebra satisfied by canonical
variables describing noncommutative spaces. The large number of possible free
parameters in these calculations is reduced to a manageable amount by imposing
various different versions of PT-symmetry on the underlying spaces, which are
dictated by the specific physical problem under consideration. The
representations for the corresponding operators are in general non-Hermitian
with regard to standard inner products and obey algebras whose uncertainty
relations lead to minimal length, areas or volumes in phase space. We analyze
in particular one three dimensional solution which may be decomposed to a two
dimensional noncommutative space plus one commuting space component and also
into a one dimensional noncommutative space plus two commuting space
components. We study some explicit models on these type of noncommutative
spaces.Comment: 18 page