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DI microconference


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June 19th, 2007, 9.30 a.m.:

Analytic and algebraic methods III

Villa Lanna, Prague

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Speakers


Tamas Fulop (Institute of Nuclear Physics, Rez)

Uwe Guenther (Research Center Dresden-Rossendorf, Germany)

Maurice Kibler (Institute of Nuclear Physics, University of Lyon)

Giuseppe Scolarici (Universita di Lecce and INFN)

Gerald V. Dunne (University of Connecticut)

Mark Harmer (Institute of Nuclear Physics, Rez)

Piergiulio Tempesta (Scuola Normale Superiore, Pisa)

Frederick G. Scholtz (University of Stellenbosch)

Stefan Rauch-Wojciechowski (Linkoping University)


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Lectures, with timetable in pdf




Tamas Fulop: Singular potentials in quantum mechanics and ambiguity in the self-adjoint Hamiltonian

Uwe Guenther: Projective Hilbert space structures at exceptional points

Maurice Kibler: Bases (MUBs) and operators (SIC-POVMs) used in quantum information

Giuseppe Scolarici: The complex projection of quasianti-Hermitian quaternionic dynamics

Gerald V. Dunne: Functional Determinants in Quantum Field Theory

Mark Harmer: Spin Filtering and the Rashba Hamiltonian

Piergiulio Tempesta: Topological field theories and integrable hierarchies

Frederik G. Scholtz: The Why, If and But of Quasi-hermitian Quantum Mechanics

Stefan Rauch-Wojciechowski Algorithmic criterion of separability - a solution of an old Jacobi problem

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Abstracts



Tamas Fulop:
For many singular potentials, including the Coulomb potential and V(x) = g/x^2 with g in a certain range, the Schrödinger Hamiltonian is not automatically self-adjoint. Such operators admit more than one self-adjoint domain, and the spectrum and all physical consequences depend seriously on the self-adjoint version chosen. The talk wishes to discuss how to identify the self-adjoint domains via boundary conditions and what physical differences emerge for different self-adjoint versions of the Hamiltonian.


Uwe Guenther:
A non-Hermitian complex symmetric 2x2 matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase jump behavior are analyzed and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT-symmetrically extended Quantum Mechanics are discussed.
The talk is partially based on arXiv:0704.1291v2 [math-ph].


Maurice Kibler:
This lecture is devoted to MUBs (mutually unbiased bases) and SIC-POVMs (positive operator valued measures which are symmetric informationally complete) used in quantum information. A polar decomposition of the (nondeformed) Lie algebra of SU(2) is generated from a q-deformed oscillator algebra. This leads to a nonstandard representation theory of SU(2). More precisely, the standard scheme {j2,jz}, where j2 is the Casimir operator and jz the Cartan generator of SU(2), is replaced by a scheme {j2,vra}, where the operator vra is defined in the enveloping algebra of the Lie algebra su(2). It is shown that the eigenvectors of the commuting set {j2,vra} generate MUBs. A similar approach for generating SIC-POVMs is briefly described.

Giuseppe Scolarici:
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.

Gerald V. Dunne:
Functional determinants play an important role in quantum field theory: for example, in the computation of effective actions, in lattice gauge theory, and in semiclassical approximations. These determinants contain both perturbative and non-perturbative information, but are difficult to compute except in special cases. I review what is known about this problem and describe an approach to higher dimensional cases where there is a symmetry that makes the problem separable. This represents a generalization to higher dimensions of the Gel'fand-Yaglom theorem for determinants of one-dimensional operators.

Mark Harmer:
We consider a quantum graph consisting of a ring with Rashba hamiltonian and an arbitrary number of semi-infinite wires attached. We describe the scattering matrix for this system and investigate spin filtering for a three terminal device.

Piergiulio Tempesta:
The connection between topological field theories, Frobenius manifolds and integrable hierarchies of PDEs is discussed. The example of the Toda systems is illustrated. (joint work with B. Dubrovin).

Frederik G. Scholtz:
A selective overview of the history and motivation for quasi-hermitian quantum mechanics. The lecture also offers some outlooks on possible future developments.

Stefan Rauch-Wojciechowski:
About 1842 C.G.J.Jacobi developed the method of solving natural Hamiltonian systems H=(1/2)p^2+V(q) through separation of variables in the related Hamilton-Jacobi(H-J) equation. It consists of finding new, suitable variables x(q) such that the H-J equation admits a separated solution. C.G.J.Jacobi stated also that there is no systematic way of finding whether separation variables exist. Surprisingly the C.G.J.Jacobi view has been incorrect. I have found an algorithmic criterion of separability that for any given potential V(q) determines whether separation coordinates exist and in such case determines them in maximally n-steps. This criterion is also useful for solving the Schrödinger equation since the stationary Schrödinger equation is separable in the same orthogonal coordinates as the H-J equation of a natural Hamiltonian.

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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;

free lunch in Villa: deadline elapsed;

lunch in Villa: may still be ordered by email, is to be paid on the spot in the morning;

accommodation in Prague: may be ordered by email; caution: the capacity is almost filled;

free accommodation in Rez: may be ordered by email; caution: the capacity is almost filled.


The currently updated list of participants is here (in pdf)


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FAQs:


social programme and similar services

not provided

conference fee:

zero

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Proceedings

written versions of some contributions were refereed
and published in the electronic journal "SIGMA"

available now on the following address.

thematically related manuscripts:
may still be submitted by email (znojil@ujf.cas.cz)
in the form compatible with the Journal's conditions

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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



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