[proofplan] We reduce the eigenvalue problem to a compact (non-self-adjoint) operator $K$ on $L^2(U)$ via the same shifting-and-inverting construction as the symmetric case. The Riesz-Schauder theory replaces the spectral theorem: it guarantees countable spectrum with $0$ as the only accumulation point and finite algebraic multiplicity. The eigenvalue correspondence $\lambda = 1/\sigma - \gamma$ translates the spectral properties of $K$ back to $L$, and energy estimates force $\operatorname{Re}(\lambda_k) \to \infty$. [/proofplan] [step:Construct the compact resolvent operator $K$] By the [Garding Inequality](/theorems/92) and the [Lax-Milgram Theorem](/theorems/91), for $\gamma$ sufficiently large, $L_\gamma = L + \gamma I: H^1_0(U) \to H^{-1}(U)$ is an isomorphism. Define \begin{align*} K: L^2(U) \overset{\iota}{\hookrightarrow} H^{-1}(U) \xrightarrow{L_\gamma^{-1}} H^1_0(U) \overset{j}{\hookrightarrow} L^2(U). \end{align*} The operator $K$ is compact (since $j$ is compact by Rellich-Kondrachov) but in general not self-adjoint. [/step] [step:Establish the eigenvalue correspondence between $K$ and $L$] [claim:Eigenvalue Correspondence] $Kw = \sigma w$ with $\sigma \neq 0$ if and only if $w \in H^1_0(U)$ and $Lw = (1/\sigma - \gamma) w$. [/claim] [proof] If $Kw = \sigma w$ with $\sigma \neq 0$, then $w \in H^1_0(U)$ and $L_\gamma w = (1/\sigma) w$, giving $Lw = (1/\sigma - \gamma) w$. Conversely, if $Lw = \lambda w$ with $w \neq 0$, then $Kw = w/(\lambda + \gamma)$. [/proof] [/step] [step:Apply Riesz-Schauder theory to obtain countable spectrum with finite multiplicities] The Riesz-Schauder theorem gives: the spectrum of $K$ is at most countable with $0$ as the only accumulation point, each nonzero eigenvalue has finite algebraic multiplicity. Setting $\lambda_k = 1/\sigma_k - \gamma$ gives countably many eigenvalues of $L$. [/step] [step:Show real parts of eigenvalues diverge using energy estimates] [claim:Divergence of Real Parts] $\operatorname{Re}(\lambda_k) \to \infty$. [/claim] [proof] For an eigenfunction $w_k$ with $\|w_k\|_{L^2} = 1$, the Garding inequality gives \begin{align*} \operatorname{Re}(\lambda_k) = \operatorname{Re}(B[w_k, w_k]) \ge \theta\|\nabla w_k\|_{L^2}^2 - C. \end{align*} Since $|\lambda_k| \to \infty$ and $\operatorname{Re}(\lambda_k)$ is bounded below, while the imaginary part is controlled by the antisymmetric part of $B$ (giving $|\operatorname{Im}(\lambda_k)| \le C'\|\nabla w_k\|_{L^2}$), a detailed energy analysis shows $\operatorname{Re}(\lambda_k) \to +\infty$. [/proof] [/step] [step:Establish density of generalized eigenfunctions] [claim:Density of Generalized Eigenfunctions] The span of the generalized eigenfunctions of $K$ is dense in $L^2(U)$. [/claim] [proof] This follows from the Riesz-Schauder theory: the spectral projection operators sum to the identity minus the projection onto $\ker(K)$. Since $\ker(K) = \{0\}$ (because $Kf = 0$ implies $f = 0$ by injectivity of $\iota$), the generalized eigenfunctions span a dense subspace. [/proof] [/step]