Uwe Guenther:
After a brief exposition of the Naimark dilated PT brachistochrone [PRL
101, 230404 (2008)] evidence is provided that the dilation (doubling of
the Hilbert space dimension) preserves the brachistochrone features of
the model. The dilated PT brachistochrone in 4D-Hilbert space behaves as
an effective Hermitian brachistochrone in the 2D subspace spanned by the
4D initial and final states.
Andreas Fring:
We study a lattice version of the Yang-Lee model which is characterized
by a non-Hermitian quantum spin chain Hamiltonian. We analyse the role
played by PT-symmetry in order to guarantee the reality of the
spectrum in certain regions of values of the coupling constants
and find the Hermitian counterpart of the Hamiltonian for small
values of the number of sites, both exactly and perturbatively.
Finally we compute the magnetization of the chain.
Emanuela Caliceti:
In the framework of perturbation theory criteria for the
reality and non-reality of the spectrum of PT-symmetric Schroedinger
operators have been recently established. After describing the main
criteria and their applications, including cases of discrete spectra
and of continuous ones as well, the mathematical techniques
supporting the proofs of the results are outlined.
Boris Shapiro:
I present some recent results on the root distribution of
eigenfunctions in the univariate case.
In particular, it will be explained that for the classical quartic
oscillator all these roots are either real or pure imaginary. I will
also describe that for an arbitrary polynomial potential these roots
(after an appropriate scaling) asymptotically fill an interesting
part of the Stokes line for a standard potential depending only on
the leading term of the original potential when the absolute value of
the eigenvalue tends to infinity.
Recommended preparatory reading: papers with A. Gabrielov and A. Eremenko, e.g.,
``High energy eigenfunctions of one-dimensional Schroedinger operators with polynomial potentials" [Comput. Methods Funct Theory 8(2), (2008), 513-529.)] or ``Zeros of eigenfunctions of some anharmonic oscillators" [Annales de l'institut Fourier, 58(2), (2008), 603-624].
Vincenzo Grecchi: Abstract:``The prove the conjecture of Bender and Weniger about the Pade'
summability of the perturbation series of each eigenvalue of the cubic
oscillator, is given and discussed."
Geza Levai: The asymptotic region of potentials have strong impact on their
general properties. This problem is especially interesting for
PT-symmetric potentials, the real and imaginary components of
which allow for a wider variety of asymptotic properties than
in the case of purely real potentials. We consider exactly solvable
potentials defined on an infinite domain and investigate their
scattering and bound states with special attention to the boundary
conditions determined by the asymptotic regions. The examples
include potentials with asymptotically vanishing and non-vanishing
real and imaginary potential components (Scarf II, Rosen-Morse II,
Coulomb, etc.).
Stefan Rauch-Wojciechowski:
Abstract in pdf
transcribed also in plain text, imperfectly, as follows:
The classical separability theory of potential Newton equations q&& = -ÑV (q) and of the related
natural Hamiltonians ( ) 2
2
H = 1 p +V q has been a cornerstone of almost all exactly solved problems in
Analytical Mechanics and a pivotal factor in building early theory of quantisation in Quantum
Mechanics. This theory is well summarised in recent papers by Benenti, Chanu, Rastelli in JMP (2002,
2003).
A natural generalisation of this theory are (discovered in Linköping 1999) systems of quasipotential
Newton equations of the form ( ) ( ) ( ) 1 q = M q = -A q Ñk q - && , n qÎR , A(q) -Killing matrix.
If q&& = M(q) admit two quadratic integrals of motion then there are n quadratic integrals of motion
and the equations are completely integrable. These Newton equations are then characterised through a
certain Poisson pencil and, equivalently, through a system of ( 1) 2
1 n n - 2nd order PDE´s - the
Fundamental Equations, which for potential forces reduce to the well-known Bertrand-Darboux
equations. We have also shown that bi-quasipotential Newton equations are separable in new types of
coordinates given by nonconfocal quadric surfaces.
The theory of bi-quasipotential Newton equations have been soon generalised by Sarlet and
Crampin (2001) to the framework of Riemannian manifolds as geodesic equations with a forcing term.
In 2005 S.Benenti discovered that the bi-quasipotential property of Newton equations leads to the
Levi-Civita dynamically equivalent systems on Riemannian manifolds.
I shall review main theorems of theory of quasipotential Newton equations and will talk about an
interesting subclass of driven Newton equations y M ( y)
&& = , x M ( y, x) V ( y, x) x
= = -Ñ
¯
&& for
which knowledge of a single quadratic integral E q cofGq k(q) = & t & + , n q = ( y, x)Î R is sufficient for
separability of the time dependent Hamilton-Jacobi equation corresponding to Newton equations of
the form x V ( y(t), x) x
&& = -Ñ .
For the subclass of triangular systems of Newton equations ( ,..., ) k k 1 k q&& = M q q , k = 1,...,n even
a stronger (1 n) theorem is valid. It says that knowledge of one quadratic integral implies existence
of n quadratic integrals and the system is solvable by separation of variables. The emerging
separation coordinates are described for n = 2 and for n = 3 .
Hugh Jones:
In the context of quasi-Hermitian theories we address the problem of
how functional integrals and Feynman diagrams ``know" about the
metric $\eta$. The resolution is that, although $\eta$ does not
appear explicitly,
the derivation of the path integral and Feynman rules is based on the
Heisenberg equations of motion, and these only take their standard
form when matrix elements are evaluated using $\eta$.
Daniel Hook:
We postulate the form of the probability amplitude $\rho(z)$ for a PT quantum
mechanical system. As an illustrative example, we calculate $\rho(z)$ for a number
of the eigenstates of the harmonic oscillator system and present a numerical study
surrounding these results.
Roberto Tateo: We discuss a three-parameter family of PT -symmetric Hamiltonians, show
that real
eigenvalues merge and become complex at quadratic and cubic exceptional
points.
The mapping of the phase diagram is completed using a combination of
numerical, analytical and perturbative approaches.
(With P.Dorey, C.Dunning and A.Lishman)
Steven Duplij:
"We consider an analog of Legendre transform for non-convex functions
with vanishing Hessian and propose to mix the envelope and general
solutions of the Clairaut equation. Then we show that the procedure of
finding a Hamiltonian for a singular Lagrangian is just that of solving
a corresponding Clairaut equation with a subsequent application of the
proposed Legendre-Clairaut transformation. We do not use
the Lagrange multiplier method and show the origin of
the Dirac primary constraints in the presented framework."
Petr Siegl:
Spectra of the second derivative operators corresponding to the PT -symmetric
point interactions on a line are studied. The particular PT -symmetric point
interactions causing unusual spectral effects are investigated for the systems
defined on finite interval as well. The spectrum of this type of interactions is very
far from the self-adjoint case despite of PT -symmetry, P-pseudo-Hermiticity
and T -self-adjointness.
Giuseppe Scolarici:
We discuss bi-hamiltonian quantum descriptions when composite
systems and interaction among them are considered. Some examples
are also exhibited.
Takuya Mine: We consider the Schr\"odinger operators
in the three-dimensional space
with magnetic fields supported in concentric tori,
which are generated by toroidal solenoids.
We prove that the operators converge to an operator
in the norm resolvent sense
as the thicknesses of the tori tend to 0,
if we choose the gauge of the vector potentials appropriately.
The limit operator is the Schr\"odinger operator
with a singular magnetic field supported on a circle.
This is a collaborated work with A. Iwatsuka and S. Shimada.
Hynek Bila:
Elementary analysis of scattering in
non-Hermitian field theories will be
presented on the toy model with imaginary cubic interaction. Necessary
modifications of the standard perturbative approach demanded by the
crypto-hermiticity of the theory will be discussed.
Miloslav Znojil:
Complete list of eligible metrics (i.e., of physical inner products in Hilbert
space of states) is derived for the
one-parametric family of
cryptohermitian toy Hamiltonians of paper I (M. Znojil, Phys. Rev. D
78 (2008) 025026).
A natural classification of these metrics is found and interpreted
as a fundamental length $\theta$. The asymptotically local inner product of paper I
recurs at minimal
$\theta=0$ while the popular ${\cal CPT}-$symmetric option
appears to corresponds to
the maximal
$\theta \to \infty$.