Aktas,
Metin
Supersymmetric Target
Potentials of Effective Mass Schrödinger Equations for PT-/Non-PT-Symmetric and Non-Hermitian
Morse Potential
We construct the supersymmetric target potentials of position dependent
effective mass Schrödinger equation for PT-/Non-PT-Symmetric and
non-Hermitian Morse potential by choosing three mass distributions. The
energy spectra of the bound states and corresponding wavefunctions for
the potential are given in the exact closed forms. |
Alavi,
Ali (poster)
Statistical mechanics of
pseudo Hermitian systems in equilibrium
To study quantum mechanical systems composed of indistinguishable
entities, as most physical systems are, one finds that it is advisable
to rewrite the ensemble theory in a language that is more natural to a
quantum-mechanical treatment, namely the language of the operators and
the wave functions. Once we set out to study these systems in detail,
we encounter a stream of new and altogether different physical
concepts. In particular, we find that the behavior of even a
noninteracting system, such as an ideal gas, departs considerably from
the pattern set by the so-called classical treatments. In the presence
of interactions the pattern becomes still more complicated. Recently a
large amount of research work has been devoted to the study of
Mathematics and Physics of pseudo Hermitian operators. In this paper we
first study the formulation of quantum statistics of pseudo Hermitian
system in equilibrium. Then in the framework of pseudo Hermitian
systems we study the statistics of various ensembles including the
microcanonical and canonical ensembles. We also consider for
illustration the statistical mechanics of a toy model. Thermodynamics
of such systems can be easily derived using partition functions. |
Algin,
Abdullah (pdf)
(q1,q2)-deformed SUSY
algebra for SUq1/q2(n)-invariant bosons
We study the system of n ordinary fermions and n (q1,q2)-deformed
bosons with SUq1/q2(n)-symmetry. We particularly discuss the
two-dimensional case in detail. Using the Fock space representation of
the system, we construct a (q1,q2)-deformed SUSY algebra. The total
deformed Hamiltonian and the energy levels for the system are also
obtained. Some physical applications of the two-parameter deformed
quantum group invariant bosonic oscillators are mentioned. |
Andrianov,
Alexander (pdf)
Dual oscillators with PT
symmetry in path integral approach
The nonlinear transformation in the QM path integral for partition
function will be outlined and used to establish the equivalence between
PT symmetric oscillators
with anharmonicity and some other systems like
Quantum Pendulum. |
Arabshahi,
Hadi (poster)
Mass dependence of hermitian
perturbative framework
An algorithm for constructing a whole class of mass dependence of
hermitian perturbative framework has been carried out. The method is
applied to the Hermitian analogue of the PT-symmetric cubic anharmonic
oscillator. |
Bagchi,
Bijan Kumar
Bound states in
position-dependent mass problems
We consider the specific models of Zhu-Kroemer and BenDaniel-Duke in a
sech2-mass background and point out interesting
correspondences withe the stationary 1-soliton and 2-soliton solutions
of the KdV equation in a supersymmetric framework. We also explore the
relevance of our scheme with the recently noticed su(1,1) realization
of Swanson's non-Hermitian Hamiltonian. |
Behrndt,
Jussi (pdf)
Dissipative scattering
systems
Quantum systems which interact with their environment are often modeled
by maximal dissipative operators or so-called Pseudo-Hamiltonians. In
this talk the scattering theory for such open systems is considered. In
particular it will be shown how the scattering matrix of a dissipative
scattering system can be expressed in terms of an abstract
Titchmarsh-Weyl function. The results are applied to a class of
Sturm-Liouville operators arising in dissipative and quantum
transmitting Schrödinger-Poisson systems. |
Bender,
Carl
Faster Than Hermitian
Quantum Mechanics
Given an initial quantum state |I> and a final quantum state |F>
in a Hilbert space, there exist Hamiltonians H under which |I>
evolves into |F>. Consider the following quantum brachistochrone
problem: Subject to the constraint that the difference between the
largest and smallest eigenvalues of H is held fixed, which H achieves
this transformation in the least time τ? For Hermitian Hamiltonians τ
has a nonzero lower bound. However, with non-Hermitian PT-symmetric
Hamiltonians satisfying the same energy constraint, τ can be made
arbitrarily small without violating the time-energy uncertainty
principle. This is because for such Hamiltonians the path from |I>
to |F> can be made short. The mechanism described here is similar to
that in general, relativity in which the distance between two
space-time points can be made small if they are connected by a
wormhole. This result may have applications in quantum computing. |
Bentaiba,
Mustapha (poster)
The effective potential and
Resummation procedure to multidimentional complex cubic potentials for
weak and strong-coupling
The method for the recursive calculation of the effective potential is
applied successfully in case of weakcoupling limit (g→0) to a
multidimensional complex cubic potential. In strong-coupling limit
(g→∞), the result is resumed using the variational perturbation theory
(VPT). It is found that the convergence of VPT-results approaches those
expected. |
Bila, Hynek (pdf)
Complexification of energies
in simple relativistic systems
The parametric dependence of the eigenvalues is dicussed in
comparison to non-relativistic models. The Coulomb and square-well
potentials are discussed within the Dirac and Klein-Gordon equation,
with
special attention paid to the behaviour near the exceptional points. |
Caliceti,
Emanuela (pdf)
PT-symmetric non-selfadjoint
operators, diagonalizable and non- diagonalizable, with real discrete
spectrum.
Two natural mathematical questions arising in this
context of PT-symmetry are
(i) the determination of conditions under
which PT-symmetry actually
yields real spectrum and (ii) the
examination of whether or not this phenomenon can still be
understood in terms of selfadjoint spectral theory; for example, it
has been remarked that if a PT-symmetric
Schrödinger operator with
real spectrum is diagonalizable, then it is conjugate to a
selfadjoint operator through a similarity map. Hence the question
arises whether PT-symmetric
Schroedinger-type operators with real
spectrum are always diagonalizable.
First we give an explicit example of a PT-symmetric Schroedinger-type
operator with purely real and discrete spectrum, which cannot be
diagonalized because of the occurence of Jordan blocks. Second we
identify a new class of non-selfadjoint PT-symmetric operators with
purely real spectrum. |
Chaharsough
Shirazi, Atoosa (poster)
Linear covariant quantum
gravity in de Sitter space-time
The recent observational data are strongly in favor of a positive
acceleration of the present universe. Therefore, in a first
approximation, the back ground space-time might be considered as a de
Sitter space-time. In usual quantization of the traceless rank-2 mass
less tensor field(linear quantum gravity) in de Sitter space two
problems appear. The first is that the theory is not fully covariant,
and the second that graviton propagator in the linear approximation has
a infrared divergence for largely separated points. The main ingredient
for constructing linear covariant quantum gravity is the presence of
two different types of negative norm states. This
construction,indefinite metric quantization, allows us to avoid the
infrared divergence, and enables us to obtain a covariant two-point
function. |
Curtright,
Thomas (pdf)
Quasi-hermitian Liouville
Theory
I will briefly discuss properties of quasi-hermitian (PT-symmetric) theories in the
context of a simple exactly solvable example, "imaginary" Liouville theory. I
will use a deformation quantization approach, i.e. QM in phase space.
Then I will discuss the field theory extension of this model, of
interest in string theory. |
Debnath,
Swapna (poster)
On Isospectral Partners for PT-symmetric Potentials
We discuss Darboux method to obtain isospectral partners of complex PT-symmetric potentials.
We have observed that the supersymmetric
partner potential has the same spectrum, including the zero energy
eigenstate. |
Dorey,
Patrick
PT symmetry and symmetry
breaking for some simple
inhomogeneous potentials
We explore the pattern of PT symmetry breaking for
a class of models related via the ODE/IM correspondence
to the Perk-Schultz models of statistical mechanics.
The structure is surprisingly rich, with exceptional
points corresponding to the joining of two and of three
levels being seen. |
Fasihi
Aghbolagh, Mohammad Ali (poster)
Two dimensional pseudo
hermitian quasi exactly solvable models
In this paper two dimensional pseudo Hermitian quasi exactly solvable
models have been studied. Using SU(3) group coherent states and their
corresponding D-algebras, we obtained one dimensional quasi exactly
solvable Schrödinger operator. Also the gauge transformation and
change
of variables were used to reduce the Schrödinger operator to
Hamiltonian form. Moreover considering the matrix form of the
Hamiltonian and imposing pseudo Hermitian conditions, and regarding the
normalizability condition on the wave functions, we classified the two
dimensional pseudo Hermitian quasi exactly solvable models . |
Fatemi,
Sara (pdf)
Conformally invariant
''photon'' field in de Sitter universe
Recent astrophysical data indicated that our universe might currently
be in a de Sitter phase. In a first approximation, the background
space-time that we live, might be considered as a de Sitter space-time.
If this is so, it is important to formulate quantum field theory in de
Sitter space-time. In this paper conformally invariant wave equation in
de Sitter space-time for vector “photon” field is considered, solution
and related two-point function in ambient space notation, have been
calculated. The Hilbert space structure and field operator in terms of
coordinate independent de Sitter plane waves have been defined. |
Feinberg,
Joshua
PT- Symmetric Quantum
Mechanical Matrix Models
We define and study PT-symmetric
quantum mechanical matrix models, in
the large-N limit. We focus on the U(N)-singlet and adjoint sectors. In
particular, the case of inverted quartic matrix motential is studied,
with special emphasis on the parity anomaly and its manifestation in
the dynamics of matrix eigenvalues. |
Fring,
Andreas (pdf)
A master non-Hermitian cubic
PT-symmetric 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 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 -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
from below
unbounded -x6-potential. As the latter models are closely
related to the
complex cubic potential they require a perturbative treatment.
This is joint work with Paulo Assis. |
Ganguly,
Asish (pdf)
PT-symmetric potential with
position-dependent mass
Using standard supersymmetric method one can obtain two partner
Hamiltonians with same position-dependent mass exhibiting identical
spectra. We have extended this procedure by allowing the superpotential
to be a complex function. A specific relation between the
superpotential and mass is derived by forcing one partner potential to
be strictly real. Provided the underlying relation holds, a class of
complex potentials can be generated with real spectra. An example has
been constructed, where the target potential is PT-symmetric for the chosen mass
function. |
Geyer,
Hendrik (pdf)
The metric in
quasi-Hermitian quantum mechanics: overview and recent results
We trace the introduction and role of the metric in the formalism of
quasi-Hermitian quantum mechanics, including its role in the
identification of observables compatible with a given
non-Hermitian/quasi-Hermitian Hamiltonian.
The construction of the metric, systematically obtained from a Moyal
product analysis, and various other aspects of the formalism, are
illustrated, primarily in the context of the non-Hermitian oscillator.
Amongst other results, this illustrates how the metric dependence
manifests itself on the level of an equivalent Hermitian Hamiltonian
and sheds some light on the metric dependence (or not) of its classical
limit. |
Graefe,
Eva-Maria (pdf)
A non-Hermitian two-mode
Bose-Hubbard system
We study an N-particle, two-mode Bose-Hubbard system, modelling a
Bose-Einstein condensate
in a double-well potential. Furthermore we introduce an effective
non-Hermiticity to the model to describe a coupling to a continuum from
one of the two wells. The resulting eigenvalues are in general complex
where the imaginary part (resonance width) describes the rate with wich
an eigenstate decays to the continuum. In dependence on the systems
parameters the resonance widths undergo a sequence of bifurcations,
which are studied in more detail. In the hermitian case important
features of the manyparticle system can be understood introducing a
meanfield-approximation which is often called 'classical'. Accordingly
we investigate an analogue approximation for the non-Hermitian system. |
Guenther,
Uwe (pdf)
Projective Hilbert space
structures near exceptional points and the quantum brachistochrone
The talk consists of two parts. In the first part, a brief overview
of projective Hilbert space structures related to exceptional points
(EPs) is presented. The apparent contradiction between operator
(matrix) perturbation schemes related to root-vector expansions and
expansions in terms of eigenvectors for diagonal spectral
decompositions is projectively resolved. In the second part of the
talk, the gained insight is used for a geometric analysis of the
brachistochrone problem for non-Hermitian as well as for PT-symmetric/pseudo-Hermitian
quantum mechanical systems. The
passage time contraction for non-Hermitian Hamiltonians compared to
Hermitian ones is attributed to a distance contraction in projective
Hilbert space due to non-unitary evolution. In the limiting case
when a parameter dependent Hamiltonian approaches an EP in its
spectral decomposition the distance between the coalescing
eigenvectors vanishes and with it the passage time of the
brachistochrone. |
Hajizadeh,
Kobra (pdf)
Indefinite metric
quantization
In this paper some properties of the indefinite metric Fock
quantization are studied. It is shown that the presence of negative
norm states or negative frequency solutions is indispensable for a
fully covariant quantization of the minimally coupled scalar field in
de Sitter space. Finally the λφ4 theory in Minkowski space
is considered by the use of this theory. |
Han,
Shiang Yi (pdf)
An intrinsical complex
Hamiltonian in the Schrödinger equation
Many varieties of complex Hamiltonians have been considered in the
literature, and conditions ensuring the reality of their eigenvalues
have been investigated. The aim of this paper is to show that the
existence of such complex Hamiltonians is due to a universal property
that every quantum system is accompanied by an intrinsic complex
Hamiltonian in such a way that the equation describing this quantum
system is just an expression for the total energy conservation of the
accompanying intrinsic complex Hamiltonian. It is revealed that quantum
systems subjected to different complex potentials, having totally
different real or imaginary eigenvalues, may possess similar intrinsic
complex Hamiltonians. A typical Hermitian system represented by H = p2
+ x2, a PT-symmetric
system
H = p2 + x2 + i x , and a non-PT-symmetric system
H = p2 + x2 + i x -x are shown to possess
identical intrinsic complex Hamiltonian, identical eigen-functions, and
identical quantum trajectory under linear coordinate translation in
complex plane. |
Heiss,
Dieter
Exceptional points for three
coalescing levels
The coalescence of three levels has particular attractive features.
Even though it may be difficult to realise such event in the laboratory
(three additional real parameters must be adjusted), to take up the
challenge seems worthwhile. In the same way as the chiral behaviour
of a usual EP can give a direction on a line, the state vectors in
the vicinity of an EP3 provide an orientation in the plane. The
distinction between left and right handedness depends on the
distribution of the widths of the three levels in the vicinity of
the point of coalescence. |
Hook,
Daniel (pdf)
Complex Trajectories of a
Simple Pendulum
The trajectory described by the simple classical pendulum is a well
known secondary school calculation. However, we may ask the question
"What happens if we start the motion of the pendulum bob outside one of
the turning points of the motion?" In this case the trajectory followed
by the bob lies beyond the real line in the complex plane. We note that
the Hamiltonian governing this system is PT symmetric in nature and
study the trajectories of the bob under a number of different
conditions. |
Jones,
Hugh (pdf)
Coupling the Hermitian and
pseudo-Hermitian worlds
We show with a number of examples that it is possible to couple
Hermitian and pseudo-Hermitian Hamiltonians together, maintaining a
real spectrum, provided that the coupling is not too strong. When the
coupling exceeds a critical value the energy levels of the coupled
system become complex. |
Kerimov,
Gul-Mirza (pdf)
Lie algebraic approach to
non-Hermitian Hamiltonians with real spectra
An algebraic technique, useful in studying of a non-Hermitian
Hamiltonians with real spectra, is presented. The method is illustrated
by explicit application to a family of one-dimensional potentials. |
Klaus,
Martin (pdf)
Spectral properties of
non-Hermitian systems arising in fiber optics
We are concerned with the spectral properties of the Zakharov-Shabat
system. This is a system of two coupled first-order linear differential
equations that plays an important role in the study of pulse
propagation in optical fibers. Associated with this system is a
non-Hermitian eigenvalue problem
which for physically realistic pulse shapes exhibits nonreal
eigenvalues. We discuss recent work towards a rigorous understanding of
why complex eigenvalues arise,
where they are located, and how they move as certain parameters vary.
Included are results on eigenvalue crossings, in particular their
location and dependence on the shape of the potential. |
Krejcirik,
David (pdf)
Bound states in a PT-symmetric waveguide
We introduce a planar waveguide of constant width with non-Hermitian PT-symmetric
heterogeneous boundary conditions and study the spectrum
of this system. Under the condition that the heterogeneity is local in
a sense, we prove that the essential spectrum is real and stable, find
sufficient conditions which guarantee the existence of weakly coupled
eigenvalues and construct the leading terms of their asymptotic
expansions. This is a joint work with Denis Borisov. |
Lavagno,
Andrea
q-deformed quantum
mechanics and q-Hermitian operators
Starting on a q-deformed canonical quantization rule,
postulated on the basis of the non-commutative q-differential calculus,
we study a generalized q-deformed Schrödinger equation. Such an
equation of motion can be viewed as the quantum stochastic counterpart
of a generalized classical kinetic equation, reproducing a 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 basic
quantum mechanics assumptions. |
Lévai,
Géza (pdf)
PT symmetry in
multi-dimensional solvable quantum potentials
Since its introduction, PT
symmetric quantum mechanics has led to a
number of interesting findings, however, the vast majority of these
concerned one-dimensional systems. Recently we proposed the systematic
generalization of these investigations to higher-dimensional potentials
by the separation of the radial and angular variables [1]. It was found
that the angular variables play an essential role in introducing
non-Hermiticity in the imaginary potential terms. Here we discuss
conditions under which solvable potentials can be constructed in 2 or
3 spatial dimensions with unbroken or spontaneously broken PT symmetry.
This requires the solution of ordinary second-order differential
equations obtained after the separation of the variables. We present
examples to illustrate various aspects of this problem.
[1] G. Lévai, J. Phys. A: Math. Theor. 40 (2007) F273 |
Liu,
Quan Hui
Constraint induced mean
curvature dependence of Cartesian momentum operators
For a particle moves on the curved smooth surface, the constraint
induced terms into the Cartesian momentum operators are the mean
curvature of the surface multiplied by the components of the unit
normal vector of the surface. When the constraint is free, the
Cartesian momentum operators reproduces their usual forms. |
Mazharimousavi,
S. Habib (poster)
η -weak-Pseudo-Hermiticity
generators and exact solvability
|
Milton,
Kimball (pdf)
PT-Symmetric Quantum Field
Theory: PTQED
A PT-Symmetric version of
Quantum Electrodynamics, which was anomaly
free, was proposed at the 1st International Workshop on
Pseudo-Hermitian Hamiltonians
in Quantum Physics [Czech. J. Phys. 54, 85 (2004)]. The C operator,
used to define the unitary norm in the theory, was constructed
perturbatively the following year [Phys. Lett. B 613, 97 (2005)]. Last
year it was
demonstrated that the perturbation theory for PTQED is in fact that
obtained
by the naive substitution e->ie [J. Phys. A 39, 1657 (2006)].
However, nonperturbative effects, not captured by this substitution,
must be responsible
for achieving unitarity of the S-matrix for PTQED. Recent progress in
understanding the content of PTQED will be discussed in two and four
spacetime
dimensions. |
Mustafa,
Omar (pdf)
Non-Hermitian von Roos
Hamiltonian's η -weak-pseudo-Hermiticity and exact-solvability
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.
Two "user -friendly" reference-target maps are introduced to serve for
exact-solvability of some non-Hermitian η-weak-pseudo-Hermitian
position dependent mass Hamiltonians. A non-Hermitian PT-symmetric
Scarf II and a non-Hermitian periodic-type PT-symmetric Samsonov-Roy
potentials are used as reference models in a "user-friendly"
reference-target map and the corresponding isospectral Hamiltonians are
obtained. It is observed that for each exactly-solvable reference
Hamiltonian there is a corresponding set of exactly-solvable target
Hamiltonians. |
Mostafazadeh,
Ali (pdf)
Pseudo-Hermiticity and
PT-symmetry: Mysteries, Facts, and Fiction
In this talk I will give a brief review of pseudo-Hermitian quantum
mechanics, its connection with PT-symmetry, and its correspondence with
classical mechanics. I will then discuss the subtleties involved in
dealing with time-dependent pseudo-Hermitian Hamiltonians, give a
thorough analysis of the geometry of the state space in
pseudo-Hermitian quantum mechanics, elaborate on the prospects of the
“Faster than Hermitian Quantum Mechanics” and outline a novel
application of pseudo-Hermitian quantum mechanics in classical
electrodynamics. |
Nakamura,
Yuichi (pdf)
Non-Hermitian quantum
mechanics of strongly correlated systems
In my talk, we argue that the imaginary part of zeros of the dispersion
relation of the elementary excitation of a quantum systems is equal to
the inverse correlation length. We confirm the relation for the Hubbard
model[1] in the half-filled case and the S=1/2 antiferromagnetic XXZ
chain[2]. In order to search zeros of the dispersion relation in the
complex momentum space efficiently, we introduce a non-Hermitian
generalization of quantum systems by adding an imaginary vector
potential ig to the momentum operator[3]. We calculate a non-Hermitian
critical point gc at which the energy gap between the ground state and
the excited state vanishes and above which the ground-state energy
becomes complex. We show numerical data of gc for the Heisenberg chain
with nearest- and next-nearest-neighbor interactions. We also show that
we can obtain the inverse correlation length of this model by
extrapolating the finite-size estimates of gc to infinite systems.
[1] Y. Nakamura and N. Hatano, in preparation.
[2] K. Okunishi, Y. Akutsu, N. Akutsu and T. Yamamoto, Phys. Rev. B 64
(2001) 104432.
[3] Y. Nakamura and N. Hatano, Physica B 378-380 (2006) 292; J. Phys.
Soc. Jpn. 75 (2006) 114001. |
Ogilvie,
Michael (pdf)
PT symmetry and large-N
models
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 wave functions can be
reduced to the solution 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. |
Payandeh,
Farrin (pdf)
Quantum field theory in
Krein space
Field quantization in flat space-time is according to choosing the
positive norm states which results a covariant quantization under
Poincare group,but some infinities appears in theory. But field
quantization in Krein space results a automatically renormalized
theory. |
Prvanovic,
Slobodan (poster)
The operator version of
Poisson bracket as the new Lie bracket of quantum mechanics and the
obstruction free quantization
The operator version of Poisson bracket for quantum mechanical
observables is defined. It is shown that it has all properties of the
Lie bracket and that quantum mechanical observables form the Lie
algebra with this bracket as a product. Moreover, it is shown that
operator version of Poisson bracket can substitute commutator in the
von Neumann equation. Next, the algebraic product of quantum mechanics
is defined as the ordinary multiplication followed by the application
of superoperator that orders involved operators, with the ordering rule
which coincides with Weyl ordering rule. This superoperator is defined
in a way that allows obstruction free quantization when the observables
are considered from the point of view of the algebra. |
Quesne,
Christiane
Swanson's non-Hermitian
Hamiltonian and su(1,1): a way towards generalizations
A recently constructed family of metric operators for Swanson's PT-symmetric Hamiltonian
is re-examined in the light of su(1,1). An
alternative derivation, relying only on properties of su(1,1)
generators, is proposed. Being independent of the realization
considered for the latter, it opens the way towards the construction of
generalized Swanson's non-Hermitian (not necessarily PT-symmetric)
Hamiltonians related by similarity to Hermitian ones. Some examples of
them are reviewed. |
Rajendrasinh,
Parmar
Supersymmetry in higher
dimension
|
Ren,
ShaoXu (pdf)
Intrinsic and Inherent
Orbital Angular Momentum
We find a new type of orbital angular momentum represented by
non-Hermitian operators. It is the generator of SO(3) in non-Hermitian
space. We find the eigenvalue of the orbital angular momentum of
quantum particle could be attached nonintegral and non-half-integral
quantum number in this non-Hermitian space. Further, particle orbital
angular momentum possesses minimum intrinsic and inherent non-zero
value. We Suggest to conduct experiments to verify the hypothesis of
possible existent minimum. |
Robert,
Didier
Supersymmetry vs ghosts
We consider the simplest
nontrivial supersymmetric quantum mechamical system involving higher
derivatives. We unravel the existence of additional bosonic and
fermionic integrals of motion forming a nontrivial algebra. This allows
one to obtain the exact solution both in the classical and quantum
cases. The supercharges Q and Qbar are not anymore Hermitially
conjugate to each other, which allows for the presence of negative
energies in the spectrum. We show that the spectrum of the Hamiltonian
is unbounded from below. It is discrete and infinitely degenerate in
the free oscillator-like case and becomes continuous running from -∞ to
∞ when interactions are added. Notwithstanding the absence of the
ground state, the Hamiltonian is Hermitian and the evolution operator
is unitary. The algebra involves two complex supercharges, but each
level is 3-fold rather than 4-fold degenerate. This unusual feature is
due to the fact that certain combinations of supercharges acting on the
eigenstates of the Hamiltonian bring them out of Hilbert space. This is
a joint work with A. Smilga (subatech, Nantes).
|
Rotter,
Ingrid (pdf)
Dynamics of open quantum
systems described by a non-Hermitian Hamiltonian
The eigenvalues of an
non-hermitian Hamilton operator may cross in the complex energy plane.
The crossing points are branch points with a non-trivial topological
structure. They are related to spectral reordering processes caused by
the bifurcation of the imaginary parts of the eigenvalues (decay
widths) at strong coupling between system and environment. In
approaching the crossing points, the phases of the eigenfunctions of
the non-hermitian Hamilton operator are not rigid. The phases of some
eigenfunctions align with the phases of the wave functions of the
environment. These phase variations cause dynamical effects in open
quantum systems which are observable. As an example, the transmission
through quantum dots is considered in the regime of overlapping
resonances. The transmission is strongly enhanced in a certain
(critical) range of the coupling strength between system and
environment.
|
Roy,
Barnana (pdf)
Nonlocal Variant of PT
Symmetric Potential
Factorisation approach for complex Hamiltonians is used to obtain
exactly solvable nonlocal variant of non-Hermitian PT-invariant local
potentials. Exact eigenvaluesand eigenfunctions of nonlocal
PT-invariant hyperbolic Rosen-Morse potential are obtained. |
Samsonov,
Boris (pdf)
Non-Hermitian dynamics and a
Hilbert space ``relativity principle"
Quantum mechanics (QM) is reexamined from the view-point of operator
equivalence classes. Two assumptions are used as starting points:
- Any operator with a diagonal spectral decomposition, purely
real
spectrum and which is densely defined in a suitable Hilbert space
may describe a physical observable.
- Two sets of observables with operators which are related by
a
similarity transformation (i.e. which are elements from operator
equivalence classes and lie on corresponding conjugacy orbits) are
physically indistinguishable. In other words, the physical
properties related to the observables X
and AXA-1 are
exactly
the same when A is a
non-singular bounded operator (Hilbert space
``relativity principle").
This implies that a given set of observables which is equivalent to
a set with all operators Hermitian in a corresponding Hilbert space
will not lead to the appearance of new properties compared to
conventional QM. In contrast, if a set of observables consists of
Hermitian and non-Hermitian operators in a certain Hilbert space
then new effects can be expected which will go beyond those of
conventional QM.
In this respect we analyze the evolution of a Hermitian observable
governed by a non-Hermitian Hamiltonian.
The general approach is illustrated by a toy model 2 X 2-matrix
Hamiltonian as it was recently used by Bender et al (Phys. Rev.
Lett. 98 (2007) 040403) for considerations of a PT-symmetric
quantum brachistochrone problem. We calculate the corresponding spin
flip probabilities and show that the spin flips in non-Hermitian
models may have flip (passage) times which are shorter or longer
compared to those of Hermitian systems. The effect depends on the
concrete exceptional point which is approached in the parameter
space of the model.
|
Scolari,
Giuseppe (pdf)
The complex projection of
quasianti-Hermitian quaternionic dynamics
We show that the complex projection of quantum dynamics ruled by
quasianti-Hermitian quaternionic (time-independent) Hamiltonians are
one-parameter semigroup dynamics in the space of complex
quasi-Hermitian density matrices. Some examples are also considered. |
Shalaby,
Abouzeid
Hermiticity Breaking and
Restoration in the (gφ4+hφ6)1+1 Field
Theoretic Model
We introduce hermiticity as a new symmetry and show that when starting
with a model which is Hermitian in the classical level, quantum
corrections can break hermiticity while the theory stay physically
acceptable. To show this, we calculated the effective potential of the
(gφ4+hφ6)1+1 model up to first order
in g and h couplings which is sufficient as the region of interest has
finite correlation length for which mean field calculation may suffice.
We show that, in the literature, there is a skipped phase of the theory
due to the wrong believe that the theory in the broken hermiticity
phase is unphysical. However, in view of recent discoveries of the
reality of the spectrum of the non-Hermitian but PT-symmetric theories,
in the broken hermiticity phase the theory possesses PT-symmetry and
thus physically acceptable. In fact, ignoring this phase will lead to
violation of universality when comparing this model predictions with
other models in the same class of universality. |
Shanley,
Paul (pdf)
Spectral Properties of an
Eigenvalue Problem due to Richardson
An eigenvalue problem involving a second order differential equation
was introduced by Richardson in the early years of the twentieth
century. It takes the form of an indefinite Sturm-Liouville problem and
in the years since its inception, the complicated dependence of its
eigenvalues on a parameter has defied explanation. We introduce a
transformation that maps the Richardson eigenvalues onto those of a
Schrödinger operator. The parametric dependence of the Richardson
spectrum is then understood by tracing its image in the more familiar
Schrödinger framework. Our study indicates that in some quantum
problems that exhibit PT-symmetry,
there are curves off the real and
imaginary axes of the coupling parameter on which the eigenvalues are
real. |
Shamshutdinova,
Varvara (poster)
Dynamical qubit controlling
based on
pseudosupersymmetry in two-level systems
Basing on a recently discovered non-linear pseudosupersymmetry in
two-level systems [1] we propose a new method for controlling a qubit
state. Namely, for a flux qubit 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 value of 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.
[1] Samsonov B.F., Shamshutdinova V.V., J. Phys. A: Math. Gen. 38
(2005) 4715-4725;
Shamshutdinova V.V., Samsonov B.F., Gitman D.M., Ann. Phys. 322 (2007)
1043-1061 |
Siegl,
Petr (poster)
Quasi-Hermitian Model
with Point Interactions and Supersymmetry
Models with two PT-symmetric point interactions compatible with
supersymmetry are studied. Two classes of boundary conditions providing
non-equivalent models are found. Positive, bounded metric operator is
constructed for both models. |
Sinha,
Anjana (pdf)
A class of Non Hermitian
Models with real energies
We analyze a class of non Hermitian quadratic Hamiltonians, which
are of the form H = ω A† A + α A2 + β A† 2 , where ω ,
α , β are real constants, with α ≠ β , and A† and A are generalized
creation and annihilation operators. It is shown that the
eigenenergies are real for a certain range of values of the
parameters. A similarity transformation ρ, mapping the non
Hermitian Hamiltonian H to a
Hermitian one h, is also
obtained. It is shown that H
and h share identical
energies.
As explicit examples, the solutions of a couple of models based on
the trigonometric Rosen-Morse I and the hyperbolic Rosen-Morse II
type potentials are obtained. We also study the case when the non
Hermitian Hamiltonian is PT
symmetric. |
Smilga,
Andrei (pdf)
Cryptoreality of
nonanticommutative Hamiltonians
We note that, though nonanticommutative (NAC) deformations of Minkowski
supersymmetric theories do not respect the reality condition and seem
to lead to non-Hermitian Hamiltonians H, the latter belong to the class
of ``cryptoreal'' Hamiltonians considered recently by Bender and
collaborators. They can be made manifestly Hermitian via the similarity
transformation H -> eR H e-R with a properly
chosen R. The deformed model enjoys the same supersymmetry algebra as
the undeformed one, though being realized differently on the involved
canonical variables. Besides quantummechanical models, we treat, along
similar lines, some NAC deformed field models in 4D Minkowski space. |
Sokolov,
Andrey (poster)
Index Theorem and spectral
design of non-Hermitian non-diagonalizable Hamiltonians
The Index Theorem on relation between Jordan structures of intertwined
Hamiltonians and the behavior of elements of canonical basis of
supercharge kernel at infinity is presented. This theorem can be used
for spectral design of non-diagonalizable Hamiltonians. The
illustrative example of non-Hermitian reflectionless Hamiltonian with
one Jordan cell is made. |
Tater,
Milos (pdf)
Quasi-exact oscillators and
their tobogganic bound states
In a finite segment of the bound-state energy spectrum, certain
(i.e., typically, the charged harmonic or the parity-invariant
sextic) elementary potentials V(x)
are known to generate the
exact wave functions ψn(x)
in closed form. We pay attention
to the remaining energy
levels and study their dependence on
the tobogganic winding number of their asymptotic, PT symmetric boundary conditions.
Joint work with M. Znojil. |
Tateo,
Roberto (pdf)
Pseudo-differential
equations, generalised eigenvalue problems and the Bethe Ansatz
We outline a relationship between the Bethe Ansatz and generalised
eigenvalue problems of pseudo-differential equations, and discuss its
relevance to problems in PT-symmetric
quantum mechanics, in quantum
field theory and in the geometric Langlands correspondence. |
Trunk,
Carsten (pdf)
Perturbation theory for
self-adjoint operators in Krein spaces
We present some recent results in the perturbation theory for
self-adjoint operators in Krein spaces. In the case
of an unperturbed self-adjoint operator in a Krein spaces with real
spectrum only we formulate conditions which ensure real spectrum of the
perturbed operator.
As applications we consider PT-symmetric
Hamiltonians and
Sturm-Liouville operators with an indefinite weight. |
Weston,
Robert (pdf)
PT Symmetry on the Lattice:
The Quantum Group Invariant XXZ Spin-Chain
We will discuss exact results connecting the research areas of
integrable lattice systems and non-Hermitian Hamiltonians. In
particular, we investigate the PT-symmetry
of the quantum group
invariant XXZ chain. We show that the PT-operator
commutes with the
quantum group action and also discuss the transformation properties of
the Bethe wavefunction. We exploit the fact that the Hamiltonian is an
element of the Temperley-Lieb algebra in order to give an explicit and
exact construction of an operator that ensures quasi-Hermiticity of the
model. This construction relys on earlier ideas related to quantum
group reduction. We then employ this result in connection with the
quantum analogue of Schur-Weyl duality to introduce a dual pair of
C-operators, both of which have closed algebraic expressions. |
Wu, Junde (poster)
Remark for Quantum
Observables
The set of bounded observables for a
quantum system is represented by the set of bounded self-adjoint
operators $S(H)$ on a complex Hilbert space $H$. The usual order
$A\leq B$ on $S(H)$ is determined by assuming that the expectation
of $A$ is not greater than the expectation of $B$ for every state
of the system. We may think of $\leq $ as a numerical order on
$S(H)$. Recent, Gudder introduce a new order $\preccurlyeq$ on
$S(H)$ that may be interpreted as a logical order. This new order
is determined by assuming that $A\preccurlyeq B$ if the
proposition that A has a value in $\Delta$ implies the proposition
that B has a value in $\Delta$ for every Borel set $\Delta$ not
containing 0. Moreover, this order can be generated by an orthosum
$\oplus$ that endows $S(H)$ with the structure of a generalized
orthoalgebra. In this paper, we discuss some element properties
for this new order of Quantum Observables. |
Yu, Zhijian (poster)
Operation Continuity of
Efffect Algebras
In 1936, Birkhoff and Von Neumann introduced the
famous closed subspaces lattice of a separable
infinite-dimensional Hilbert space as a quantum logic structure to
describe the quantum mechanical system entity. In 1994, Foulis and
Bennet introduced the following algebraic system $(E, \bot,
\oplus, 0, 1)$ to model unsharp quantum logics and called it the
effect algebra:
Let $E$ be a set with two special elements 0, 1, $\bot$ be a
subset of $E\times E$, if $(a, b)\in \bot$, denote $a\bot b$, let
$\oplus: \bot\rightarrow E$ be a partially binary operation, and
the following axioms hold:
(E1). (Commutative Law) If $a, b\in E$ and $a\bot b$, then $b\bot
a$ and $a\oplus b=b\oplus a$.
(E2). (Associative Law) If $a, b, c\in E, a\bot b$ and $(a\oplus
b)\bot c$, then $b\bot c, a\bot (b\oplus c)$ and $(a\oplus
b)\oplus c=a\oplus (b\oplus c)$.
(E3). (Orthocomplementation Law) For each $a\in E$ there exists a
unique $b\in E$ such that $a\bot b$ and $a\oplus b=1$.
(E4). (Zero-Unit Law) If $a\in E$ and $1\bot a$, then $a=0$.
In this paper, we prove that the operations $\oplus$ and $\ominus$
of effect algebra are continuous with respect to its ideal
topology, and if the effect algebras are the lattice effect
algebras, then under some conditions, the lattice operations
$\vee$ and $\wedge$ of lattice effect algebra are also continuous
with respect to the ideal topology. These results showed that the
ideal topology of effect algebra is a very important topology. |
Yuce, Cem (pdf)
Complex Spectrum of a
Spontaneously Unbroken PT
Symmetric Hamiltonian
It is believed that unbroken PT
symmetry is sufficient to guarantee
that the spectrum of a non-Hermitian Hamiltonian is real. It is proven
that this is not true. It is shown that the spectrum is not real for a
non-Hermitian and spontaneously unbroken Hamiltonian. |
Znojil,
Miloslav (pdf)
Physics near exceptional
points
After a brief review of the
phenomenology of models with H ≠ H† (non-relativistic as
well as relativistic, used in quantum mechanics as well as in field
theory and in non-quantum areas like MHD), specific attention will be
paid to the problem of de-freezing of relevant degrees of freedom.
Among various simple models of the related phenomenon of transition
between real and complex energies (at ``exceptional points"),
finite-dimensionalsolvable models will be analyzed in more technical
detail.
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