microseminar
informal seminar of Doppler Institute
on quantum theory and related topics and methods,
usually
on Thursdays at 10.30 a.m. in Rez,
on the territory of the
Department of Theoretical Physics
of the
Nuclear Physics Institute.
Enthusiastic visitors and/or speakers are always
welcome.
Forthcoming sessions:
| Place: | OTF seminar room (second floor) |
| Date: | |
| Time: | 10:30 a. m. (20 minutes talk + unconstrained time for discussion) |
| Speaker: |
|
| Title: | A spectral isoperimetric inequality for cones |
Abstract: | Spectral isoperimetric inequalities are one of the most famous issues in spectral geometry, the first rigorous results dating almost a century back to the papers of {Faber} and {Krahn}. Recently such inequalities appeared in the context of Schr\"odinger operators with singular potentials used as models of `leaky quantum wires' and similar systems. In particular, for the 2-D Schroedinger operator with a $\delta$-potential of a fixed strength supported on a loop of a given length it was shown by {Exner}, {Harrell} and {Loss} that its principal eigenvalue is maximal when the loop is a circle. The corresponding problem in 3-D is more involved. For closed simply connected surfaces of a fixed area the sphere gives a local maximum of the ground-state eigenvalue, however, the result does not have a global validity. Nevertheless, there are 3-D Schroedinger operators with singular interactions supported on surfaces for which one is able to derive a spectral isoperimetric inequality that holds not only locally. The aim of my talk is to discuss one such class considered in our preprint (arXiv:1512.01970). The surfaces in question are of a conical shape, both finite and infinite. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimisers for a class of energy functionals. The main novel idea consists in the special choice of the test function for the Birman-Schwiner principle. Joint work with Pavel Exner. |
Archive:
| Date and time: | Thursday, January 7th, 2016, 13:30 a.m. (irregular time) |
| Speaker: | Ondrej Turek (OTF UJF in Rez and BLTP JINR in Dubna, Russia) |
| Title and abstract: | The effect of edge lengths ratio on the spectrum of a hexagonal lattice |
We analyze the spectrum of a Laplacian operator on a dilated honeycomb lattice. The lattice is assumed to be dilated along its axis of symmetry and supporting $\delta$ potentials of strength $\alpha \neq 0$ in its vertices. It turns out that the qualitative properties of the spectrum depend on number-theoretic properties of edge lengths ratio $\theta$. We will show that the number of spectral gaps is infinite for any rational or well approximable $\theta$, whereas there are only finitely many gaps in the spectrum if $\theta$ is badly approximable and the potential strength $\alpha$ in the vertices is small compared to the Markov constant of $\theta$.
This talk is based on joint works with P. Exner. |
The archive of the older microseminars:
The list of talks during 2012 - 2015
The list of talks during 2008 - 2011
The list of the talks during 2008
A compactified list of the speakers during 2008
The list of the talks during 2007
A compactified list of the speakers during 2007
The list of the talks during 2006
A compactified list of the speakers during 2006
PS: in parallel, nested seminars of the similar type may be also sought on the webpages of our local
microconferences
devoted to the analytic an algebraic methods in physics
Info for potential/interested external speakers:
| you would be always welcome by: | all of us |
| you may choose any date and time | though Thursdays on 10.30 are preferred |
and any subject related to
|
|
| you should book your term of talk: | not later than 2 or 3 days in advance |
| your talk's length should be | 20 minutes |
| time for subsequent questions: | unlimited |
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