## 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

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

return upwards

## 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. "

return upwards

## 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&& = -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 $\theta \to \infty$.

return upwards

## 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

return upwards

not provided

zero.

return upwards

## Refereed proceedings

open also to non-participants:

## contributions should be submitted by email (znojil@ujf.cas.cz)

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

return upwards

## 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

return upwards

## 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

return upwards
May 29th, 2009, ultimate update by Miloslav Znojil                            return upwards                            jump to the webpage of DI