DI microconference
May 27 - 28, 2009, 9.30 a.m. - 5.00 p.m.
Analytic and algebraic methods V
Villa Lanna, Prague
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Speakers
Uwe Guenther (FZ Rossendorf, Germany)
Andreas Fring (City University, London, UK)
Emanuela Caliceti (Bologna University, Italy)
Boris Shapiro
(Stockholm University, Sweden)
Vincenzo Grecchi (Bologna University, Italy)
Geza Levai: (ATOMKI, Debrecen, Hungary)
Stefan Rauch-Wojciechowski (Linkoeping University, Sweden)
Hugh Jones (Imperial College, London, UK)
Daniel Hook (Imperial College, London, UK)
Roberto Tateo (Universita' di Torino, Italy)
Steven Duplij (V.N. Karazin Kharkov National University, Ukraine)
Petr Siegl (NPI & Universit\'{e} Paris 7 - Denis
Diderot)
Giuseppe Scolarici (Koc University, Istanbul, Turkey)
Takuya Mine (Kyoto Institute of Technology, Japan)
Hynek Bila (NPI Rez)
Miloslav Znojil (NPI Rez)
List of participants
is here (in pdf),
their photos (copyright: S. Duplij) are
here and here
(click here for more info)
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Titles of lectures
Uwe Guenther: The 4D Naimark dilated PT brachistochrone
as 2D Hermitian brachistochrone
Andreas Fring: The Ising quantum spin chain in an imaginary field
Emanuela Caliceti: PT-symmetric Schroedinger operators: spectral and perturbation
theory
Boris Shapiro: 'Exact and asymptotic results on root distribution of
eigenfunctions of a univariate Schroedinger equation with a
polynomial potential'
Vincenzo Grecchi:
Stieltjes property of cubic oscillator
Geza Levai: On the asymptotic properties of exactly solvable PT-symmetric potentials
Stefan Rauch-Wojciechowski:
Structure and separability of driven and triangular systems of Newton equations
Hugh Jones: "Which Green functions does the path integral represent?"
(with Ray Rivers)
Daniel Hook: "Numerical study of PT quantum mechanical systems"
Roberto Tateo: `PT symmetry breaking and exceptional points'
Steven Duplij: A novel Hamiltonian procedure for constraint theories
Petr Siegl: Surprising spectra of PT -symmetric point interactions
Giuseppe Scolarici: Bi-hamiltonian descriptions for composite quantum systems
Takuya Mine:
: Norm resolvent convergence to Schroedinger operators
with infinitesimally thin toroidal megnetic fields
"
Hynek Bila:
: Scattering in i phi^3 pseudo-scalar theory.
"
Miloslav Znojil:
: All metrics for a toy Hamiltonian.
"
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Abstracts
The book of abstracts in pdf
is here
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&& = -ŃV (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 Ńk 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
= = -Ń
Ż
&& 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
&& = -Ń .
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
$\theta \to \infty$.
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Registration
proceeded by email;
(needed due to the limited capacity of the Villa; principle: first-come, first-served)
deadline, the date when the number on the registered participants gets equal to 32: not reached,
info:
the number of participants was 19.
meals:
(two) lunches in Villa: ordered by email;
paid,
in cash, on the spot (100 CZK each);
accommodation:
people accommodated in Villa (+420 224 321 278):
1. Uwe Guenther, Dresden, Germany, 1 night: 27/28 May 2009
2. Andreas Fring, London, 1 night: 27/28 May 2009
3. Vincenzo Grecchi, Bologna, 3 nights: 26/27, 27/28, 28/29
4. Emanuela Caliceti, Bologna, 3 nights: 26/27, 27/28, 28/29
5. Giuseppe Scolarici, Istanbul, 3 nights: 26/27, 27/28, 28/29
6. Hugh Jones, London, 3 nights: 26/27, 27/28, 28/29
7. Boris Shapiro, Stockholm, Sweden, 2 nights: 26/27, 27//28
8. Roberto Tateo, University of Torino, Italy, 3 nights: 26/27, 27/28, 28/29
9. Geza Levai, Debrecen, 1 night: 27/28
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FAQs:
social programme and similar services
not provided
conference fee:
zero.
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Refereed proceedings
open also to non-participants:
accepted MSs will be published in a
dedicated issue of
"SIGMA",
= electronic journal with specific advantages:
top-quality referees,
unlimited number of pages,
visibility and accessibility (arXiv overlay).
deadline for submission of manuscripts: September 30, 2009
can individually be postponed by email
Manuscripts
the form of MSs must be
compatible
with the Journal's conditions
the style advice available through the following
address.
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contacts
e-mail:
znojil@ujf.cas.cz
letter:
Miloslav Znojil
Nuclear Physics Institute,250 68 Rez ,Czech Republic
FAX:
+420 2 20940165
phone:
+420 2 6617 3286 or +420 724 747 898
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OTHER MEETINGS IN PRAGUE:
Quantization day 2
March 24, 2009
Selected Topics in Mathematical and Particle Physics
May 5 - 7, 2009
Integrable Systems and
Quantum Symmetries
June 18 - 20, 2009
XVI International Congress on Mathematical Physics:
August 3 - 8, 2009
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May 29th, 2009, ultimate update by
Miloslav Znojil
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