%<<<<<<<<< Verfasser: Hagen Neidhardt
%<<<<<<<<< Betr.: Vortrag in Prag
%<<<<<<<<< Datum: 05.01.2001
\documentclass[a4paper,12pt]{article}
\title{One self-adjoint and dissipative
Schr\"{o}dinger-Poisson systems}

\author{Christoph Kaiser, \underline{Hagen Neidhardt}, Joachim Rehberg}

\date{Weierstrass-Institute for Applied Analysis and Stochastics\\[5mm]
Berlin\\[5mm]
05.01.2001}

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\newcommand{\dom}{{\mbox{\rm dom}}}


\begin{document}
\maketitle

The talk deals with the Schr\"{o}dinger-Poisson system on a finite interval $[a,b]$
of the real axis. On the Hilbert space $L^2([a,b])$ the Schr\"{o}dinger operator
$H$,
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\begin{equation}\label{1}
(H(V)f)(x) = -\frac{d^2}{dx^2}f(x) + V(x)f(x), \quad f \in \dom(H) = W^{2,2}([a,b]),
\end{equation}
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with a real potential $V(\cdot) \in L^2([a,b])$ is considered which is
supplemented either by self-adjoint (Dirichlet or Neuman) or by
dissipative boundary conditions. In both cases it is possible to
perform a career density $n(H(V))(x)$. It is assumed that the potential
$V$ satisfies the Poisson equation
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\begin{equation}\label{2}
-\frac{d^2}{dx^2}V(x) = n(H(V))(x), \quad V(a) = V_a \quad \mbox{and}
\quad V(b) = V_b
\end{equation}
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where $V_a$ and $V_b$ are given voltages. The non-linear system
consisting of equation (\ref{1}) and (\ref{2}) is called the
Schr\"{o}dinger-Poisson system. The goal is to show that
such a Schr\"{o}dinger-Poisson system has always a solution for
suitable potential classes. The problem arises from semi-conductor physics.

\end{document}