This licentiate thesis  concerns second order elliptic operators with coefficients, that is, differential operators of the form

\begin{equation}

    L u     = -\frac{1}{ \rho } \operatorname{div} A \nabla u \, , 

\end{equation}

where the function $ u : \Omega \to \mathbb{C} $ satisfies a Neumann or a Dirichlet condition at the boundary $ \partial \Omega $. Here, $ \rho (x) $ is a positive weight function, and $ A(x) $ is an elliptic coefficient matrix. Note that the resulting operator is self-adjoint precisely when $ A(x) $ is a hermitian matrix for all $ x \in \Omega $. Such operators arise in many areas of the natural sciences, and have thus been extensively studied. 

In the first paper of this thesis, we are interested in the Dirichlet and the Neumann spectra of $ L $ in the self-adjoint case. When the domain $ \Omega $ is bounded, both spectra consist of infinitely many non-negative eigenvalues $ \lambda_n $ resp. $ \mu_n $, and we are interested in inequalities of the type $ \mu_{n+k} \leq \lambda_n $ (with $ k \in \mathbb{N} $ fixed), valid for $ n=1 $ or for all $ n \in \mathbb{N} $. For the Laplacian $ -\Delta $ many inequalities of this form are known, and our aim is to generalize these results to elliptic operators under adequate assumptions on the coefficients.

In the second paper, we study the square root $ \sqrt{L} $ (and more precisely, its domain of definition $ \operatorname{dom} (L) \subseteq L^2 ( \Omega ) $) of the operator $ L $, which is defined via functional calculus. Kato's square root conjecture, formulated in the 1960s and resolved in the early 2000s, states that $ \operatorname{dom} ( \sqrt{L} ) $ coincides with the domain of the sesquilinear form associated to $ L $ (namely $ H^1_0 ( \Omega ) $ in the case of a Dirichlet boundary condition). Note that this statement is almost trivial when $ L $ is self-adjoint, but the question is considerably harder when the coefficients are not symmetric. In this paper, taking inspiration from the proof on $ \Omega = \mathbb{R}^d $, we give a new proof of this result on domains $ \Omega \subseteq \mathbb{R}^d $ that relies on a \emph{second order} approach. This method avoids the \emph{first order} framework of Dirac operators previously used for treating boundary conditions.