I am trying to solve \begin{equation} -\Delta{u} = \lambda \cdot u \end{equation} with zero-boundary-conditions on the unbounded domain $\Omega = \mathbb{R} \times (0,\pi)$ for $\lambda \ge 0$ in the Sobolev spaces $H^{1}_{0}(\Omega)$. Since the embedding of $H^{1}_{0}(\Omega) \rightarrow L^{2}(\Omega)$ is not compact, the laplacian shouldn´t have a discrete spectrum. However, I was not able to find any eigenfunctions. $u_{\lambda}(x)=sin(\sqrt{(\lambda/2}) \; x) \; sin(\sqrt{(\lambda/2)} \; y)$ is obviously not in $L^{2}(\Omega)$. But how can I prove that
1) there are no other $H^{1}_{0}$-eigenfunctions?
2) the spectrum is $[0,\infty)$?
Edit: Also, any reference on either (1) or (2) would be greatly appreciated