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 winter semester of 2026/2027 the course NTMF025 runs on Thursdays at 15.40 in "Kvasnica Lecture Room", Institute of Theoretical Physics, Troja Campus
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: resonance lifetimes,
- 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: September 29, 2026