Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 23 min ago
PreprintCC BY 4.0arXiv

Resolvent Expansions at All Spectral Thresholds for Three-Dimensional Discrete Schrödinger Operators

  • Yanhong Chen
  • Sisi Huang
  • Xiaohua Yao
Open at sourcePDFDOICite

Abstract

This paper studies resolvent expansions at all four spectral thresholds for the discrete Schrödinger operator $H=-Δ+V $ on Z3, where Δ denotes the discrete Laplacian on ℤ3 and V(n) is a real-valued decaying potential of ℤ3. At the thresholds 0, 4, 8, and 12, we first give a complete classification of the threshold types and derive the corresponding resolvent expansions in weighted ℓ2-spaces, including the non-generic cases of resonances and eigenvalues. As a result, we establish the Jensen-Kato type time-decay estimates of the evolution eitH Pac(H) to discrete Schrödinger equations with thresholds singularities on Z3, with decay O(t−3/2) when all thresholds are regular and O(t−1/2) otherwise. These results provide a three-dimensional lattice analogue of the Jensen--Kato classical analysis at zero energy threshold on ℝ3.

Related

  1. Resolvent bounds and eigenvalue estimates of generalized Schrödinger operators with complex potentials on compact manifolds
    Eduard Stefanescu
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  2. On asymptotic expansions of the density of states for Poisson distributed random Schrödinger operators
    Jan Fischer, David Hasler, Jannis Koberstein
    arXivPreprintCC BY-NC-SA 4.0Full textMathematical physicsarXiv
  3. Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on ℤ2
    Shirshendu Ganguly, Wencai Liu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  4. Optimal bounds for embedded eigenvalues of one-dimensional discrete Schrödinger operators with decaying potentials
    Jifeng Chu, Wencai Liu, Kang Lyu
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsDynamical systemsarXiv
  5. Poisson statistics for one-dimensional Anderson--Bernoulli Schrödinger operators
    Yi Han
    arXivPreprintCC BY 4.0Full textMathematical physicsProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{chen2026resolvent,
  title = {{Resolvent Expansions at All Spectral Thresholds for Three-Dimensional Discrete Schrödinger Operators}},
  author = {Yanhong Chen and Sisi Huang and Xiaohua Yao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19845},
  eprint = {2610.19845},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19845},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Resolvent Expansions at All Spectral Thresholds for Three-Dimensional Discrete Schrödinger Operators
AU  - Yanhong Chen
AU  - Sisi Huang
AU  - Xiaohua Yao
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19845
UR  - https://arxiv.org/abs/2610.19845
AB  - This paper studies resolvent expansions at all four spectral thresholds for the discrete Schrödinger operator $H=-Δ+V $ on Z^3, where Δ denotes the discrete Laplacian on ℤ^3 and V(n) is a real-valued decaying potential of ℤ^3. At the thresholds 0, 4, 8, and 12, we first give a complete classification of the threshold types and derive the corresponding resolvent expansions in weighted ℓ^2-spaces, including the non-generic cases of resonances and eigenvalues. As a result, we establish the Jensen-Kato type time-decay estimates of the evolution e^itH P_ac(H) to discrete Schrödinger equations with thresholds singularities on Z^3, with decay O(t^−3/2) when all thresholds are regular and O(t^−1/2) otherwise. These results provide a three-dimensional lattice analogue of the Jensen--Kato classical analysis at zero energy threshold on ℝ^3.
ER  -