No Spectral Invisibility for Dissipative Barrier Truncations in Any Dimension
The dissipative barrier method suppresses spectral pollution, but whether it can itself conceal genuine spectral points has remained open. Known as the graveyard problem in computational spectral theory, the higher-dimensional case has remained unresolved for more than a decade. We resolve it for Schr\"{o}dinger operators in dimensions $d\geq2$; together with the known one-dimensional theorem, this settles the no-invisibility problem in all dimensions. Let $A=-\Delta+V$ be a Dirichlet Schr\"odinger operator on a connected open set $\Omega\subseteq\mathbb R^d$ with $V\in L^1_{\mathrm{loc}}(\Omega)$ bounded below, and set $H=A+iS$, where $S\geq0$ and $S\in L^p(\Omega)$, with $1 0}$ be a nested family of nonempty connected bounded open sets such that $\Omega_R\nearrow\Omega$ as $R\nearrow+\infty$. Denote by $H_R$ the Dirichlet truncation of $H$ to $\Omega_R$. We prove that every spectral point of $H$ is detected by the truncations: for every $\lambda\in\sigma(H)$ and every neighborhood $U$ of $\lambda$, $\sigma(H_R)\cap U\neq\emptyset$ for sufficiently large $R$. Equivalently, $\sigma(H)\subseteq\liminf_{R\to\infty}\sigma(H_R)$. Thus the barrier method does not trade suppression of spectral pollution for spectral invisibility. No regularity of $\partial\Omega$ is required, and the assumptions reach the critical Sobolev scale. The proof combines compactness of the dissipative form perturbation, Cwikel-type Schatten estimates for Birman--Schwinger operators, generalized strong resolvent convergence, and a reverse Hansmann--Weyl spectral-variation inequality due to Gil'. A two-dimensional numerical example illustrates the absence of spectral invisibility for the truncated dissipative operators.