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' WannierStark
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 WannierStark systems, i.e., the KronigPenney
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?
 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
BetheSommerfeld 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). Does the discrete spectrum survive a
strong magnetic field?
 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
Their contents, LaTeXformatted, you can find 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 2017/2018 the course CTUPPKT runs on Mondays from 9.30 and Wednesdays from 11.30 in the lecture rooms T014 and T112, respectively, in the Department of Mathematics, Trojanova 13 building.
Supervised PhD theses:

Michal Jex:
Strongly singular Schroedinger operators with interactions supported by curves and surfaces,
defended in Prague, Czech Technical U., October 2016
currently a posdoc at FIZ Karlsruhe

Jiri Lipovsky:
Schroedinger operators on metric graphs,
defended in Prague, Charles U., September 2011
currently a researcher at the University of Hradec Králové

Ondrej Turek:
Schroedinger operators on metric graphs,
cotutelle de these, cosupervised by
Pierre Duclos,
defended in Prague, Czech Technical U., December 2009
currently a an assistant professor at the Ostrava University

Martin Fraas:
Models of quantum systems with strongly singular interactions,
defended in Prague, Charles U., June 2008
currently an assistant professor at Virginia Tech

Katerina Oanová (Nĕmcová):
Solvable models of quantum systems with a nontrivial geometry,
defended in Prague, Charles U., June 2004
currently in a private software company

Hynek Kovarik:
Magnetic transport on twodimensional electron systems,
defended in Prague, Charles U., September 2001
currently an associate professor at the University of Brescia

David
Krejcirik: Guides d'ondes quantiques bidimensionnels,
cotutelle de these, cosupervised by
Pierre Duclos, defended in Toulon, UTV, September 2001
currently a lecturer at the Czech Technical University

Milos Tater:
Open mesoscopic systems,
defended in Prague, NPI, October 1996
currently a researcher in the Department of Theoretical Physics
Supervised postdocs:
Supervised master theses:

Michal Jex:
Geometrically induced properties of the ground state of quantumgraph Hamiltonians,
defended in Prague, Czech Technical U., June 2012

Jiri Lipovsky:
Spectral and resonance properties of quantum graphs,
defended in Prague, Charles U., June 2008

Ondrej 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

Katerina Oanová (Nĕmcová) :
Quantum mechanics of layers with point perturbations,
defended in Prague, Charles U., May 2000

David
Krejcirik: Geometrically induced spectral and
scattering properties of quantum layers,
defended in Prague, Charles U., June 1998

David Vanek:
A double waveguide with a lateral coupling,
defended in Prague, Czech Technical U., June 1997

Elena Seresova:
Point perturbations in mesoscopic systems,
defended in Prague, CTU, June 1995
Supervised bachelor students:

Stepan Timr: Spectra of comb graphs
working on the research topic
Spectra of comb graphs,
defended in Prague, Czech Technical U., June 2011

Michal Jex:
Geometrically induced properties of the ground state of pointinteraction Hamiltonians,
defended in Prague, Czech Technical U., August 2010

Martin Vana:
Perturbations of bound states in broken waveguides,
defended in Prague, Charles U., June 2009

Jan Svarcbach:
Bound states in a of conic layer,
defended in Prague, Charles U., June 2007

Jiri Lipovsky:
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 firsthand 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 AharonovBohm flux,
 Pauli resonances in a strong magnetic field,
 Probability current vortex lines in 3D crystal models,
 Finite WannierStark 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 email to
(my surname)(at)ujf.cas.cz.
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Last update: March 4, 2018