No Spectral Invisibility for Dissipative Barrier Truncations in Any Dimension

Abstract: 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 Schrödinger operators in dimensions $d\geq2$; together with the known one-dimensional theorem, this settles the no-invisibility problem in all dimensions. Let $A=-Δ+V$ be a Dirichlet Schrödinger operator on a connected open set $Ω\subseteq\mathbb R<sup>d$ with $V\in L<sup>1_{\mathrm{loc}}(Ω)$ bounded below, and set $H=A+iS$, where $S\geq0$ and $S\in L<sup>p(Ω)$, with $1&lt;p&lt;\infty$ for $d=2$ and $d/2\leq p&lt;\infty$ for $d\geq3$. Let $$(Ω_R)_{R\&gt;0}$$ be a nested family of nonempty connected bounded open sets such that $Ω<em>R\nearrowΩ$ as $R\nearrow+\infty$. Denote by $H_R$ the Dirichlet truncation of $H$ to $Ω_R$. We prove that every spectral point of $H$ is detected by the truncations: for every $λ\inσ(H)$ and every neighborhood $U$ of $λ$, $σ(H_R)\cap U\neq\emptyset$ for sufficiently large $R$. Equivalently, $σ(H)\subseteq\liminf</em>{R\to\infty}σ(H_R)$. Thus the barrier method does not trade suppression of spectral pollution for spectral invisibility. No regularity of $\partialΩ$ 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.