Asymptotic and numerical analysis of multi-target DtN coupling

Investigate the asymptotic decay of the spectral coupling coefficients $c_{nm}(p)$, determine the convergence rate of finite-dimensional truncations, characterize the limit of nearby or touching targets, and develop efficient numerical inversion methods with respect to $p$.

Background

The paper develops a block Dirichlet-to-Neumann (DtN) formulation for encounter propagators with multiple targets. In the two-target case, the diagonal DtN blocks govern local-time evolution on each target, while the off-diagonal block and its spectral coefficients $c_{nm}(p)$ encode inter-target transfers. For positively separated targets, the off-diagonal block is smoothing and compact, which suggests that its spectral coefficients may decay and that low-rank or finite-dimensional approximations may be effective, but the paper does not establish the decay law or truncation error.

The authors also identify unresolved analytical and computational issues when targets approach one another or touch, since the positive-separation assumptions used in the convergence and smoothing arguments may then fail. In addition, the construction is carried out primarily in the Laplace variable $p$, leaving efficient numerical inversion with respect to $p$ as an explicitly stated question.