## Speakers

Andreas Fring (City University, London)

David Krejcirik and Miloslav Znojil (Institute of Nuclear Physics, Rez)

Hugh Jones (Imperial College, London)

Petr Ambroz, Zuzana Masakova, Lubomira Balkova and Severin Posta, (Czech Technical University, Prague)

plus, perhaps, another guest speaker (click and send your application here, please)

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## Lectures, with timetable in pdf

David Krejcirik: A simple PT-symmetric model

Hugh Jones: The disappearing metric operator

Miloslav Znojil: The domains of quasi-Hermiticity for certain anharmonicities

Andreas Fring: PT-symmetry and integrability

Severin Posta: Structure of the Enveloping Algebras - Some Examples

Lubomira Balkova: Combinatorial and Arithmetical Properties of In¯nite Words Associated with Quadratic Non-simple Parry Numbers

Petr Ambroz: Defects of fixed points of substitutions

Zuzana Masakova: On a class of infinite words with affine factor complexity

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## Abstracts

David Krejcirik
We introduce and study a very simple, exactly solvable PT-symmetric non-Hermitian model with purely real spectrum

Miloslav Znojil:
For a matrix family of non-Hermitian PT-symmetric models we describe the domains of free parameters where the spectra remain real

Hugh Jones:
We show that, whereas the metric operator plays an essential role in the Schroedinger formulation of non-Hermitian quantum mechanics, it does not appear explicitly in the expressions for expectation values in the functional integral formulation. Instead, the relation to the equivalent Hermitian theory is encoded in the dependence of the vacuum generating functional on the external source.

Andreas Fring:
We address the question of how integrable models may be deformed in a complex, but PT symmetric manner. We consider non-Hermitian but PT-symmetric extensions of Calogero models, which have been proposed first by Basu-Mallick and Kundu for two types of Lie algebras. We show that these extensions are meaningful for all remaining Lie algebras (Coxeter groups) and that in addition one may also extend these models beyond the rational case to trigonometric, hyperbolic and elliptic potentials. We find that all these new models remain integrable, albeit for the non-rational potentials one requires additional terms in the extension in order to maintain the integrability. In addition we propose a new family of complex PT-symmetric extensions of the Korteweg-de Vries equation. The deformed equations can be associated to a sequence of non-Hermitian Hamiltonians. The first charges related to the conservation of mass, momentum and energy are constructed. We investigate solitary wave solutions of the equation of motion for various boundary conditions. We provide a relation between the PT-symmetric extensions of Calogero models and KdV equation.
old, shorter version of the abstract:
We propose a new family of complex PT-symmetric extensions of the Korteweg-de Vries equation. The deformed equations can be associated to a sequence of non-Hermitian Hamiltonians. The first charges related to the conservation of mass, momentum and energy are constructed. We investigate solitary wave solutions of the equation of motion for various boundary conditions.

Severin Posta:
We will study the structure of enveloping algebra and quantum groups. For the algebras U(B_2) and U(A_3) we give the decomposition of ad representation on the irreducible components and the explicit form of the highest weigts vectors. We will formulated open questions.

Lubomira Balkova:
We study some arithmetical and combinatorial properties of ¯-integers for ¯ being the larger root of the equation x2 = mx¡n; m; n 2 N;m ¸ n+2 ¸ 3. We determine with the accuracy of §1 the maximal number of ¯-fractional positions, which may arise as a result of addition of two ¯-integers. For the in¯nite word u¯ coding distances between the consecutive ¯-integers, we determine precisely also the balance. The word u¯ is the only ¯xed point of the morphism A ! Am¡1B and B ! Am¡n¡1B. In the case n = 1, the corresponding in¯nite word u¯ is sturmian, and, therefore, 1-balanced. On the simplest non-sturmian example with n ¸ 2, we illustrate how closely the balance and the arithmetical properties of ¯-integers are related.
(same in pdf)

Petr Ambroz:
It is known that every finite word w contains at most |w|+1 different palindromes (the empty word being considered too). The difference between |w|+1 and the actual number of palindromes in w is called defect of w. A finite word with zero defect is called full; an infinite word is said to be full if all its prefixes are full. We prove that fixed points of several well-known substitutions are full.

Zuzana Masakova:
We consider the fixed point of a primitive substitution canonically defined by a $\beta$-numeration system. The  problem of determination of the factor complexity of such an infinite word has been solved only partially. Here we provide a necessary and sufficient condition on the Renyi expansion of 1 for having an affine factor complexity map $\C(n)$, that is, such that $\C(n)=an+b$ for any $n \in \N$.

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## Registration

proceeds by email;

principle: first-come, first-served basis:

reason: the number of participants is limited to cca 30 people

lunch in Villa: may be ordered by email before March 31st (and paid in cash on the spot in the morning)

accommodation: may be ordered by email before February 27th (presumably in Villa, to be paid in cash on the spot in the morning)

The currently updated list of participants can be found here (in pdf)

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not provided

zero

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## Proceedings

Acta Polytechnica

manuscripts:

should be prepared in Word (though Latex will also be tolerated);
should be emailed to this address (the earlier the better)

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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

February 23rd, 2007, updated by Miloslav Znojil                            return upwards                            return to the webpage of microseminars