Continuity of the spectrum for the Witten Laplacian with mixed boundary conditions

Establish the continuity of the spectrum of the operator −H, where H = σΔ − U(x) with U(x) = |∇V(x)|^2/σ − (ΔV)/2, under mixed boundary conditions consisting of zero Dirichlet conditions on Γ_A and zero Neumann conditions on Γ_R, with respect to variations of the domain Ω (for example, via exhausting or shrinking domains). This is required to complete the rigorous extension of the pointwise dual relaxation argument for lower bounds to the mixed-boundary case.

Background

The paper develops lower bounds on the principal eigenvalue using a pointwise dual relaxation (PDR) method, proving convergence in the purely Dirichlet case. Extending this proof to mixed boundary conditions (Dirichlet on Γ_A and Neumann on Γ_R) requires a continuity result for the spectrum of the symmetrized operator (the Witten Laplacian) with respect to domain variations.

The authors outline how the Dirichlet argument could generalize if one could justify domain continuity of the spectrum for mixed boundary conditions, but note that they could not locate general results establishing this property. Addressing this gap would allow the theoretical guarantees (equality of the PDR bound with the principal eigenvalue) to hold in the mixed-boundary setting.