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\noindent{\bf Mathematisches Oberseminar} {\it PDG und
Spektraltheorie} (SoSe 2016).

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\noindent{\bf Date:} 21.04.2016.

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

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\noindent{\bf Speaker:} Marco Baffetti (LMU).

\noindent{\bf Titel:} {\it The proof of the Fundamental Gap Conjecture.}


\noindent{\bf Abstract:}

\noindent In 2011 Ben Andrews and Julie Clutterbuck proved the
following Fundamental Gap Conjecture, which had been suggested for the
first time by M. van den Berg in 1983:  

\begin{theorem}[Fundamental Gap Conjecture]
Let $\Omega \subset \mathbb{R}^n$ be a bounded convex domain with
diameter $D$ and $V: \overline{\Omega} \rightarrow \mathbb{R}$ a
convex potential. Then the first two eigenvalues $\lambda_0$ and
$\lambda_1$ of the Schr\"odinger operator $-\Delta +V$ on $\Omega$
with Dirichlet boundary condition satisfy: 
\begin{equation*}
\lambda_1 - \lambda_0 \geq \dfrac{3\pi^2}{D^2}.
\end{equation*}
\end{theorem}

The crucial result proved by Andrews and Clutterbuck is a
log-concavity estimate for the first eigenfunction, which implies a
comparison theorem for the fundamental gap. The proof involves a study
of moduli of concavity for positive solutions to a parabolic
Schr\"odinger equation and of moduli of continuity for solutions to
Neumann heat equations with drift. In order to simplify some of the
arguments given by Andrews and Clutterbuck, Yue He in 2013 gave an
alternative proof of the theorem which uses different methods,
although it preserves the structure of the original proof. He employs
elliptic and parabolic maximum principles, along with geometrical
properties of convex domains. In my talk I will discuss the
Fundamental Gap Conjecture and present an outline of the two proofs.  


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

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