2026/07/24 by Matthew J. Colbrook, Marco Marletta
#math.NA #cs.NA #math-ph #math.MP #math.SP
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≥2; 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⊆\mathbb Rd with V∈ L1loc(Ω) bounded below, and set H=A+iS, where S≥0 and S∈ Lp(Ω), with 1<p<∞ for d=2 and d/2≤ p<∞ for d≥3. Let (ΩR)R>0 be a nested family of nonempty connected bounded open sets such that ΩR\nearrowΩ as R\nearrow+∞. Denote by HR the Dirichlet truncation of H to ΩR. We prove that every spectral point of H is detected by the truncations: for every λ∈σ(H) and every neighborhood U of λ, σ(HR)∩ U≠∅ for sufficiently large R. Equivalently, σ(H)⊆\liminfR→∞σ(HR). Thus the barrier method does not trade suppression of spectral pollution for spectral invisibility. No regularity of ∂Ω 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.