I am looking for a reference that could help me with the following two questions:
Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Consider a sequence of differential operators $\mathcal{L}_n: D(\mathcal{L}_n) \rightarrow L^2(\Omega)$ defined by $$\mathcal{L}_n u = -\text{div}(\kappa_n \nabla u),$$ where the coefficients $\kappa_n: \Omega \rightarrow \mathbb{R}$ are at least integrable functions that are bounded by positive constants. Now suppose the sequence $\{\kappa_n\}_{n \in \mathbb{N}}$ converges in some sense (for example strongly in $L^p(\Omega)$ for a $p \geq 1$) to a coefficient $\kappa: \Omega \rightarrow \mathbb{R}$ as $n \rightarrow \infty$.
(i) Is there an analytical expression for the eigenvalues and eigenfunctions of $\mathcal{L_n}$?
(ii) Now consider the operator $\mathcal{L}: D(\mathcal{L}) \rightarrow L^2(\Omega)$ defined by $$\mathcal{L} u = -\text{div}(\kappa \nabla u).$$ Since $\kappa_n \rightarrow \kappa$ as $n \rightarrow \infty$, is there any chance, that also the eigenvalues/eigenfunctions of $\mathcal{L}_n$ converge to the eigenvalues/eigenfunctions of $\mathcal{L}$ as $n \rightarrow \infty$ ?
Or in other words, what assumptions and which kind of convergence would we need for the coefficients $\kappa_n$ in order to see that the spectrum of $\mathcal{L}_n$ converges to the spectrum of $\mathcal{L}$?
