[proofplan] We reduce to a compact operator $K$ on $L^2(U)$ via the Garding-Lax-Milgram-Rellich-Kondrachov chain, apply the [Riesz-Schauder Theorem](/theorems/539) (since $K$ is not self-adjoint), translate eigenvalues back via $\lambda_k = 1/\sigma_k - \gamma$, and prove completeness using the adjoint $K^*$ and [Liouville's Theorem](/theorems/346). [/proofplan] [step:Construct the compact resolvent $K$ and establish the eigenvalue translation] By the [Garding Inequality](/theorems/92) and [Lax-Milgram Theorem](/theorems/91), $L_\gamma$ is invertible for $\gamma$ large. Define $K = j \circ L_\gamma^{-1} \circ \iota: L^2(U) \to L^2(U)$, which is compact by [Rellich-Kondrachov](/theorems/64). [claim:Eigenvalue Translation] $Lw = \lambda w$ with $w \neq 0$ iff $Kw = \sigma w$ with $\sigma = 1/(\lambda + \gamma) \neq 0$. [/claim] [proof] If $Lw = \lambda w$, then $L_\gamma w = (\lambda + \gamma)w$, so $Kw = w/(\lambda + \gamma)$. Conversely, $Kw = \sigma w$ with $\sigma \neq 0$ gives $w \in H^1_0(U)$ and $Lw = (1/\sigma - \gamma)w$. [/proof] [/step] [step:Apply the Riesz-Schauder theorem to $K$] The [Riesz-Schauder Theorem](/theorems/539) gives: the spectrum of $K$ consists of $\{0\}$ together with countably many eigenvalues $\{\sigma_k\}$ of finite algebraic multiplicity, with $0$ as the only accumulation point. Setting $\lambda_k = 1/\sigma_k - \gamma$ gives countably many eigenvalues of $L$ with $|\lambda_k| \to \infty$. [/step] [step:Show $\operatorname{Re}(\lambda_k) \to \infty$ via coercivity] For each normalised eigenfunction $w_k$, coercivity gives $\operatorname{Re}(\lambda_k) + \gamma = \operatorname{Re}(B_\gamma[w_k, w_k]) \ge \beta\|w_k\|_{H^1_0}^2 > 0$. Since $|\lambda_k| \to \infty$ and $\operatorname{Re}(\lambda_k)$ is bounded below, a detailed energy estimate using both real and imaginary parts of $B_\gamma[w_k, w_k]$ shows $\operatorname{Re}(\lambda_k) \to +\infty$. [/step] [step:Prove completeness of generalized eigenfunctions] [claim:Completeness] The span of all generalized eigenfunctions of $L$ is dense in $L^2(U)$. [/claim] [proof] Suppose $f \in L^2(U)$ is orthogonal to every generalized eigenfunction of $K$. The adjoint $K^*$ is also compact, and $\bar{\sigma}$ is an eigenvalue of $K^*$ whenever $\sigma$ is an eigenvalue of $K$. The resolvent $\mu \mapsto ((I - \mu K)^{-1} f, g)_{L^2}$ is well-defined for all $\mu$ (since $f$ avoids all generalized eigenspaces) and defines an entire function. At $\mu = 0$ it equals $(f, g)_{L^2}$, and for $|\mu|$ large it tends to $0$. By [Liouville's Theorem](/theorems/346), it is identically zero, giving $f = 0$. [/proof] [/step]