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\noindent{\bf Mathematisches Oberseminar} {\it PDG und
Spektraltheorie} (WiSe 2016/17).

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\noindent{\bf Date:} 02.02.2017.

\noindent{\bf Time and place:} 14:15 in B 134.

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\noindent{\bf Speaker:} Daniel Teetz (LMU).

\noindent{\bf Titel:} {\it Spectral phase transitions and non-semiclassical behaviour.}

\noindent{\bf Abstract:} Quantum systems depending continuously on
some parameter $\lambda$ may exhibit a sudden change in their
behaviour along the variation of $\lambda$. This phenomenon is what we
call a spectral phase transition. 
We study a system that was introduced by Barseghyan and Exner in 2012
which features a potential unbounded from below with infinite phase
space volume below any energy. 
The potential depends linearly on a coupling parameter $\lambda$,
where a spectral phase transition happens at a certain critical value
of $\lambda$.\\ 
We discuss recent results by Barseghyan et al.: In the supercritical
case, any energy is allowed. This is the expected behaviour from
semiclassical analysis, since the classical phase space volume below
any energy is infinite.  
In contrast to this, the energy levels are purely discrete and bounded
below for subcritical values of $\lambda$. In this case, the system
also admits Lieb-Thirring-type eigenvalue bounds, although the
semiclassical approximation still yields infinite values. 

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Thomas {\O}stergaard S{\o}rensen

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