| Abstract: |
| For an $N$-body system of linear Schr\odinger equation with space dependent interaction
between particles, one would expect that the corresponding one body equation, arising as a
mean field approximation, would have a space dependent nonlinearity. With such motivation we consider
the following model of a nonlinear reduced Schr\odinger equation with space dependent nonlinearity
\begin{align*}
-\varphi``+V(x)h`(|\varphi|^2)\varphi = \lambda \varphi,
\end{align*}
where $V(x)=-\chi_{[-1,1]} (x)$ is minus the characteristic function of the interval $[-1,1]$ and where
$h`$ is any continuous strictly increasing function. In this talk, for any negative value of $\lambda$
we present the construction and analysis of the infinitely many solutions of this equation, which are
localized in space and hence correspond to bound-states of the associated time-dependent
version of the equation. |
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