{
  "abstract": "We develop an encounter-based formulation of restricted diffusion in a bounded domain with multiple targets, resolving separately the boundary local time accumulated on each target. Unlike the single-target problem, the target projection operators do not generally commute with the governing Dirichlet-to-Neumann (DtN) operator, so that its eigenbasis does not diagonalize the local-time dependence. We overcome this difficulty by decomposing the DtN operator into target-restricted blocks. Multi-dimensional Laplace inversion then yields a convergent switching expansion, whose successive terms describe alternating diffusive transfers between the targets. We also derive an equivalent operator-valued renewal equation. For two targets, diagonalizing the target-restricted blocks provides an explicit spectral representation in terms of two DtN block spectra and an inter-target coupling matrix. When the coupling preserves spectral modes, the switching series can be resummed exactly in terms of modified Bessel functions. In general, as the off-diagonal DtN block is smoothing for separated targets, one can resort to low-rank finite-dimensional approximations. We examine an effective two-mode reduction for small, well-separated targets and identify a regime in which repeated inter-target transfers are progressively suppressed. Probabilistic interpretations and implications for diffusion-controlled reactions are discussed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19995",
  "author_total": 1,
  "authors": [
    {
      "name": "Denis S. Grebenkov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2fkm425nn9575.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19995",
  "fulltext_url": null,
  "id": "2fkm425nn9575",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19995",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Differential Equations and Boundary Problems"
    ]
  ],
  "openalex": "W7213602288",
  "pdf_url": "https://arxiv.org/pdf/2609.19995",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2fkm425nn9575.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.19995",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19995"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Encounter Propagator for Multiple Targets: A Dirichlet-to-Neumann Spectral Formalism",
  "title_text": "Encounter Propagator for Multiple Targets: A Dirichlet-to-Neumann Spectral Formalism",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/2fkm425nn9575",
  "venue": "arXiv",
  "versions": []
}