L^p-Liouville properties and realizations of Schrödinger operators on incomplete Riemannian manifolds

Abstract

The aim of the paper is to highlight and standardize in a self-contained framework the connections between some properties associated with the Laplace-Beltrami operator on possibly incomplete Riemannian manifolds. We present both new and well-known characterizations of the positivity-preserving property, essential self-adjointness, and Liouville-type results for L²-functions, using the latter to establish a clear hierarchy among these properties. Furthermore, we prove the stability of the essential self-adjointness of the Laplace-Beltrami operator on incomplete Riemannian manifolds under the removal of closed (possibly non-compact) subsets from a complete manifold, subject to suitable conditions on their (generalized) Hausdorff measure. With appropriate modifications, this framework is adapted throughout the paper to the general L^p-case for p∈ (1,+∞).

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BibTeX RIS

BibTeX

@misc{bisterzo2026lpliouville,
  title = {{$L^p$-Liouville properties and realizations of Schrödinger operators on incomplete Riemannian manifolds}},
  author = {Andrea Bisterzo and Salvatore Lincastri and Giona Veronelli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29514},
  eprint = {2610.29514},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29514},
}

RIS

TY  - UNPB
TI  - L^p-Liouville properties and realizations of Schrödinger operators on incomplete Riemannian manifolds
AU  - Andrea Bisterzo
AU  - Salvatore Lincastri
AU  - Giona Veronelli
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29514
UR  - https://arxiv.org/abs/2610.29514
AB  - The aim of the paper is to highlight and standardize in a self-contained framework the connections between some properties associated with the Laplace-Beltrami operator on possibly incomplete Riemannian manifolds. We present both new and well-known characterizations of the positivity-preserving property, essential self-adjointness, and Liouville-type results for L^2-functions, using the latter to establish a clear hierarchy among these properties. Furthermore, we prove the stability of the essential self-adjointness of the Laplace-Beltrami operator on incomplete Riemannian manifolds under the removal of closed (possibly non-compact) subsets from a complete manifold, subject to suitable conditions on their (generalized) Hausdorff measure. With appropriate modifications, this framework is adapted throughout the paper to the general L^p-case for p∈ (1,+∞).
ER  -

Abstract

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