P. Exner's Bulletin Board

• 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?
• A graph approximation of a Neumann networks (see [169, 191, 205] in the list of papers). To go beyond the free (or Kirchhoff) conditions one has use properly scaled potentials. Can such approximations be constructed in a purely geometric way? Can the general approximation on graphs derived in [209] be lifted to manifolds?
• 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). 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.
  

• I read alternatively several facultative courses at the Charles and Czech Technical Universities, specifically
  
  
  • Mathematical Methods of the Quantum Theory (CU-JSF043,044),
  • Selected Topics of Mathematical Physics (CU-TMF025), and
  • Advanced Chapters of the Quantum Theory (CTU-PPKT)
  
  Their contents, LaTeX-formatted, 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 winter semester of 2009/2010 the course CU-JSF043 is given by controlled reading.

• Jiri Lipovsky working on the research topic
  Quantum graphs and their generalizations,
  since September 2008

• Michal Jex working on the research topic
  Geometrically induced properties of the ground state of point-interaction Hamiltonians,
  since October 2009

• Ondrej Turek: Schroedinger operators on metric graphs,
  defended in Prague, Czech Technical U., December 2009
• Martin Fraas: Models of quantum systems with strongly singular interactions,
  defended in Prague, Charles U., June 2008
• Katerina Nemcova: Solvable models of quantum systems with a nontrivial geometry,
  defended in Prague, Charles U., June 2004
• Hynek Kovarik: Magnetic transport on two-dimensional electron systems,
  defended in Prague, Charles U., September 2001
• David Krejcirik: Guides d'ondes quantiques bidimensionnels,
  cotutelle de these, co-supervised by P. Duclos, defended in Toulon, UTV, September 2001
• Milos Tater: Open mesoscopic systems,
  defended in Prague, NPI, October 1996

• Denis Borisov working on
  Spectral properties of perturbed quantum waveguides,
  November 2005 - April 2007

• 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 Nemcova: 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

• 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

• 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.