Quantum graphs in Mathematics, Physics and Applications

Program:
14.00--14.40 David Krejcirik
The effective Hamiltonian in curved quantum waveguides and when it does not work

The Dirichlet Laplacian in a curved three-dimensional tube built along a spatial (bounded or unbounded) curve is investigated in the limit when the uniform cross-section diminishes. Both deformations due to bending and twisting are considered. We show that the Laplacian converges in a norm resolvent sense to the well known one-dimensional Schroedinger operator whose potential is expressed in terms of the curvature of the reference curve, the twisting angle and a constant measuring the asymmetry of the cross-section.

Contrary to previous results, we allow reference curves to have non-continuous and possibly vanishing curvature. For such curves, the distinguished Frenet frame need not exist and, moreover, the known approaches to establish the result do not work. We ask the question under which minimal regularity assumptions the effective one-dimensional approximation holds.

Our main ideas how to establish the norm-resolvent convergence under the minimal regularity assumptions are to use an alternative frame defined by a parallel transport along the curve and a refined smoothing of the curvature via the Steklov approximation. On the negative side, we construct an explicit waveguide for which the usefulness of the spectral information provided by the standard effective Hamiltonian is rather doubtful.

14.50--15.30 Ram Band
Finding the nodal points on a quantum graph

The investigation of nodal patterns on manifolds has began already in the 19th century by the pioneering work of Chladni on the nodal structures of vibrating plates.
Sturm's oscillation theorem states that the n-th vibrational mode of a string has n-1 nodal points equally distributed on the string.
However, finding the location of the nodal points on a general quantum graphs is far less trivial task.
This is despite a quantum graph being nothing more than a structure of strings attached to each other.
We manage to study the number of nodal points and their location without solving the full eigenvalue problem on the graph.
This is done by defining an energy function on the space of the nodal points' possible locations and examining its critical points.

This work relates to the approach taken by Helffer, Hoffmann-Ostenhof and Terracini to study Schroedinger operators on two-dimensional domains. Quantum graphs obey the analogue of the results by Helffer et al, and are used to further generalize them.

This is a joint work with Gregory Berkolaiko, Hillel Raz and Uzy Smilansky.

15.50--16.30 Mats-Erik Pistol
Mapping a continuous band-structure model to a lattice model

We have investigated the connection between the k.p-model and the tight-binding model in solid state physics.
Both of these models are used to calculate the band-structure The k.p-model assumes the solid to be continuous wheras the tight-binding model invokes the atoms and their position.
We have found a mapping between the set of k.p-models into the set of tight-binding models such that the behaviour at small k (small wavenumbers) is the same. This solves the main problem of tight-binding models which is to find the correct parameters.
The tight-binding model can be seen as a quantum graph problem on an infinite graph using a generalised discrete laplacian.

This is a joint work with C Pryor, University of Iowa.
16.40--17.30 Delio Mugnolo
On the heat equation subject to nonlocal constraints

I will consider a heat equation on an interval subject to integral constraints on the total mass and the barycenter, instead of more common boundary conditions. The natural operator theoretical setting is that of a space of distributions on the torus.
By variational methods I will show well-posedness and some relevant spectral properties of this problem. This is joint work with Serge Nicaise (Valenciennes, France).

17.40--18.30 Petr Siegl
On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schr\"odinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated ``charge conjugation" operator, which plays the role of fundamental symmetry in a Krein space reformulation of the problem.

The talk is based on joint works with D. Krej\v ci\v r\'ik (NPI ASCR, \v Re\v z) and \mbox{J. \v Zelezn\'y} (FZU ASCR, Prague):

[1]
D. Krej\v ci\v r\'ik, P. Siegl and J. \v Zelezn\'y: \emph{On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators}, preprint available at arXiv:1108.4946.

SATURDAY, DECEMBER 10, Room 306, house 6, KRÄFTRIKET

10.00--10.40 Leszek Sirko
Isoscattering microwave networks

We discuss the scattering from a pair of isospectral microwave networks consisting of vertices connected by microwave coaxial cables. The networks are extended to scattering systems by connecting leads to infinity in a way preserving their symmetry to form isoscattering networks. We show experimentally that scattering matrices of isoscattering networks are characterized by the same polar structure. Furthermore, we demonstrate that the scattering matrices of such networks are conjugated by the transplantation relation. The results are in perfect agreement with theoretical considerations.

Acknowledgments. This work was supported by the Ministry of Science and Higher Education grant No. N N202 130239 and the European Union within European Regional Development Fund, through the grant Innovative Economy POIG.01.01.02.00-008/08. This is a joint work with Oleh Hul, Micha{\l} {\L}awniczak, Szymon Bauch, Adam Sawicki, and Marek Ku\'s.

10.50--11.30 Adam Sawicki
Scattering from isospectral quantum graphs

In 1966 Marc Kac asked 'Can one hear the shape of a drum?'. The answer was
given only in 1992, when Gordon et al. found a pair of drums with the same
spectrum. The study of isospectrality and inverse problems is obviously
not limited to drums and treats various objects such as molecules, quantum
dots and graphs. In 2005 Okada et al. conjectured that isospectral drums
can be distinguished by their scattering poles (resonances). We prove that
this is not the case for isospectral quantum graphs, i.e., isospectral
quantum graphs share the same resonance distribution. This is a joint work
with Rami Band and Uzy Smilansky.

11.50--12.30 Amru Hussein
An indefinite operator on finite metric graphs

Starting with differential expressions of the form $ -D_x {\rm sgn}\,(x) D_x$ on the real line, a generalization to metric graphs is proposed.
Their self-adjoint realizations and some of their spectral properties are discussed. These operators are motivated by a model of a cloaking
phenomenon in $\mathbb R^2.$

12.40--13.20 Jiri Lipovsky
Asympotics of resonances in quantum graphs

We study the asymptotical behaviour of the number of resonances in quantum
graphs. (More precisely, the number of poles of the resolvent contained in
the circle of diameter $R$ in the $k$ plane for large $R$.) It has been
previously shown by Davies and Pushnitski that the constant in the main
term of asymptotics is in some cases smaller than expected (these graphs
are called non-Weyl). We generalize their results for all possible
coupling conditions and find criterion under which a quantum graph has
non-Weyl behaviour. We construct unitarily equivalent Hamiltonians which
explain non-Weyl behaviour.

Furthermore, the question whether the asymptotics may change under the
influence of magnetic field is addressed. We show that the Weyl or
non-Weyl character of the asymptotics is not changed in the presence of
magnetic field. On the other hand, if corresponding non-magnetic quantum
graph is already non-Weyl, then its ``effective size'' may depend on the
value of magnetic potential.

This is a joint work with P. Exner and E.B. Davies.

14.20--15.00 Boris Shapiro
On spectral discontinuities of the classical quatric oscillator

15.10--15.50 Milos Tater
TBA

16.00--16.40 Sergey Naboko
On the Absolutely Continuous Spectrum in a Model of an Irreversible Quantum Graph

A family $ A_\alpha $ of differential operators depending on a real parameter $ \alpha \geq 0 $ is considered.
This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely
continuous spectrum $ \sigma_{a.c.} $ of the operator $ A_\alpha $ and its multiplicity for all values of the parameter.
The spectrum of $ A_0 $ is purely absolutely continuous and admits an explicit description. It turns out that
for $\alpha < \sqrt 2 $ one has $ \sigma_{a.c.} A_\alpha = \sigma_{a.c.} A_0$, including the multiplicity.
For $\alpha \ge \sqrt2$ an additional branch of the absolutely continuous spectrum arises; its source is an auxiliary
Jacobi matrix which is related to the operator $ A_\alpha $. This birth of an extra branch of the absolutely continuous spectrum is the exact mathematical expression of the effect that was interpreted by Smilansky as irreversibility.