P. Exner's Bulletin Board
Problems, Conjectures, Challenges:
The number of open problems is, of course, infinite. Here
are some I regard as worth of thinking:
- Spectra of delta' Wannier-Stark
systems (see [80,89] in the list of
papers). It is known that these systems have no absolutely continuous spectrum,
and that the spectrum is p.p. for
a "large" set of parameters values [103,111]. Can this result be extended
to all parameter values? What is the spectrum as a set for "rational"
values of the potential slope?
- A more difficult question about the spectrum of
classical Wannier-Stark systems, i.e., the Kronig-Penney
model with a linear potential. What is the spectral type and how
does the spectrum look like as a set? Is there a phase
transition between the p.p. and continuous spectrum as the
field intensity increases? Frank and Larson showed that it is not the case for rational slopes, for the irrational ones the question remains open.
- A graph approximation of a Dirichlet networks (see [195] in the list of
papers). If the overall spectral threshold is used for
energy renormalization, it understood that nontrivial limit can come from
resonances in the vicinity of the threshold. Can one work this out for
waveguides with branching?
- Band spectra of periodically perturbed quantum
waveguides due to point impurities or modification of
boundary conditions (see [94,97] in the list of papers). The number of gaps can be made
large by a suitable choice of parameters; the same is expected for periodically
curved tubes with Dirichlet boundary. Is the
Bethe-Sommerfeld conjecture stating
finiteness of the gap number nevertheless valid?
- The band spectra of periodically curved Dirichlet
waveguides (see [86] in the list of papers) are known to be
absolutely continuous for strips in the
plane. Is the same true generally in three dimensions?
- Curved quantum waveguides
in the plane have nontrivial discrete spectrum if they are
asymptotically straight. The eigenvalue shift due to a weak magnetic field
can be found by perturbation theory (see [79,86] in the list of
papers). Sometimes the discrete spectrum survives a
strong magnetic field. Is it always so?
- Various open problems concern leaky quantum graphs (see [199] in the list of papers) with respect to their spectra, various asymptotic
properties, etc.
A bulletin board serves for posting things to catch eye of
those passing by. This is why no more details about these and
related problems are given. If you find interest in some of them
and send me a message to (my surname)(at)ujf.cas.cz, a pleasure of discussing with you would be mine.
Courses:
- I read alternatively several facultative courses at the Charles and
Czech Technical Universities, specifically
You can find their contents also here in
Czech for the Charles University lectures; here and
here for the CTU course in Czech and
English, respectively.
- In the summer semester of 2023/2024 the course CTU-PPKT takes place on Fridays at 10.00 in the lecture room B-11 of the FNSPE building, Brehova 7.
Supervised PhD theses:
-
Jan Pekař:
Vertex coupling effects in spectra of quantum graphs,
Charles U., since October 2023
-
David Spitzkopf:
Soft quantum waveguides,
Charles U., since October 2022
-
Michal Jex:
Strongly singular Schroedinger operators with interactions supported by curves and surfaces,
defended in Prague, Czech Technical U., October 2016
currently posdoc at Universite Paris-Dauphine
-
Jiří Lipovský:
Quantum graphs and their generalizations,
defended in Prague, Charles U., September 2011
currently researcher at the University of Hradec Králové
-
Ondřej Turek:
Schroedinger operators on metric graphs,
cotutelle de these, co-supervised by
Pierre Duclos,
defended in Prague, Czech Technical U., December 2009
currently associate professor at the Ostrava University
-
Martin Fraas:
Models of quantum systems with strongly singular interactions,
defended in Prague, Charles U., June 2008
currently assistant professor at UC Davis
-
Kateřina Ožanová (Nĕmcová):
Solvable models of quantum systems with a nontrivial geometry,
defended in Prague, Charles U., June 2004
currently a business consultant in Stockholm
-
Hynek Kovařík:
Magnetic transport on two-dimensional electron systems,
defended in Prague, Charles U., September 2001
currently associate professor at the University of Brescia
-
David
Krejčiřík: Guides d'ondes quantiques bidimensionnels,
cotutelle de these, co-supervised by
Pierre Duclos, defended in Toulon, UTV, September 2001
currently full professor at the Czech Technical University
-
Miloš Tater:
Open mesoscopic systems,
defended in Prague, NPI, October 1996
currently researcher in the Department of Theoretical Physics
Supervised postdocs:
Supervised master theses:
-
Jiří Kolář:
Properties of broken quantum waveguides,
Czech Technical U., from October 2023
-
Ondřej Šrámek:
Magnetic effects in the spectrum of laterally coupled layers,
Czech Technical U., from October 2022
-
Jan Pekař:
Quantum graphs with circulant vertex couplings,
defended in Prague, Charles U., June 2023
-
Pavel Lokvenc:
The spectrum of periodic quantum graphs in dependence of the vertex coupling,
defended in Prague, Czech Technical U., September 2020
-
Michal Jex:
Geometrically induced properties of the ground state of quantum-graph Hamiltonians,
defended in Prague, Czech Technical U., June 2012
-
Jiří Lipovský:
Spectral and resonance properties of quantum graphs,
defended in Prague, Charles U., June 2008
-
Ondřej Turek:
Quantum graphs with strongly singular coupling at vertices,
defended in Prague, Czech Technical U., June 2006
-
Martin Fraas: Quantum systems with a generalized surface interaction,
defended in Prague, Charles U., June 2005
-
Petr Kysela:
Spectral and scattering properties of laterally coupled quantum
layers,
defended in Prague, Charles U., February 2002
-
Kateřina Ožanová (Nĕmcová):
Quantum mechanics of layers with point perturbations,
defended in Prague, Charles U., May 2000
-
David
Krejčiřík: Geometrically induced spectral and
scattering properties of quantum layers,
defended in Prague, Charles U., June 1998
-
David Vaněk:
A double waveguide with a lateral coupling,
defended in Prague, Czech Technical U., June 1997
-
Elena Šerešová:
Point perturbations in mesoscopic systems,
defended in Prague, CTU, June 1995
Supervised bachelor students:
-
Filip Breuer: Vortex curves in point-interaction systems
Czech Technical U., since October 2022
-
Adam Gottfried: Quantum systems of a mixed dimensionality
defended in Prague, Czech Technical U., September 2022
-
Michal Grňo: Magnetic transport along translationally invariant obstacles
defended in Prague, Charles U., September 2021
-
Štepán Timr: Spectra of comb graphs
defended in Prague, Czech Technical U., June 2011
-
Michal Jex:
Geometrically induced properties of the ground state of point-interaction Hamiltonians,
defended in Prague, Czech Technical U., August 2010
-
Martin Váňa:
Perturbations of bound states in broken waveguides,
defended in Prague, Charles U., June 2009
-
Jan Švarcbach:
Bound states in a of conical layer,
defended in Prague, Charles U., June 2007
-
Jiří Lipovský:
Resonances in quantum graphs,
defended in Prague, Charles U., June 2006
In addition to the problems mentioned above, there are many others.
Some of them would make a good master thesis, some can be solved with
the knowledge you have in the 3rd course. The best way to learn about
them is to attend our Quantum Circle seminar,
where students of our group, graduate and undergraduate, meet. There
you can get a first-hand contact with our activities.
While most of this stuff is nothing more than quantum mechanics,
it is related to problems of current research. To give you at least
a sample, let us mention, for instance:
- Parallel quantum waveguides with a periodic lateral coupling,
- A potential "ditch" in a magnetic field, stability
of transport with respect to perturbations,
- Quantum waveguide with an Aharonov-Bohm flux,
- Pauli resonances in a strong magnetic field,
- Probability current vortex lines in 3D crystal models,
- Finite Wannier-Stark systems with singular interactions,
a numerical study of scattering,
- Magnetic transport along a delta' line,
etc.
If you want to learn more, send me an e-mail to
(my surname)(at)ujf.cas.cz.
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Last update: February 20, 2024