Pith Number

pith:V27XMGNL

pith:2025:V27XMGNLE76H2DGQLOATUSXMOQ

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Ballistic Transport for Discrete Multi-Dimensional Schr\"odinger Operators With Decaying Potential

David Damanik (Rice University), Zhiyan Zhao (Universit\'e C\^ote d'Azur)

Discrete Schrödinger operators with potentials decaying faster than 1/|n| have purely absolutely continuous spectrum and support ballistic transport.

arxiv:2507.04988 v6 · 2025-07-07 · math-ph · math.AP · math.MP · math.SP

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Claims

C1strongest claim

We prove the absence of singular continuous spectrum for H. For the unitary evolution e^{-itH}, we prove that it exhibits ballistic transport in the sense that, for any r > 0, the weighted ℓ²-norm ||e^{-itH}u||_r grows at rate ≃ t^r as t→∞, provided that the initial state u is in the absolutely continuous subspace and satisfies ||u||_r < ∞.

C2weakest assumption

The potential satisfies V_n = o(|n|^{-1}) as |n| → ∞; this decay is invoked to apply compactness arguments and localized spectral projections that extend the free Laplacian result to the perturbed operator.

C3one line summary

Discrete Schrödinger operators on Z^d with V_n = o(|n|^{-1}) have purely absolutely continuous spectrum and exhibit ballistic transport where weighted position moments grow as t^r for AC initial states.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-04T01:09:37.801176Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

aebf7619ab27fc7d0cd05b813a4aec740ae9bc60114683e9cd23d3b103b96e88

Aliases

arxiv: 2507.04988 · arxiv_version: 2507.04988v6 · doi: 10.48550/arxiv.2507.04988 · pith_short_12: V27XMGNLE76H · pith_short_16: V27XMGNLE76H2DGQ · pith_short_8: V27XMGNL

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/V27XMGNLE76H2DGQLOATUSXMOQ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: aebf7619ab27fc7d0cd05b813a4aec740ae9bc60114683e9cd23d3b103b96e88

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "122af008fb4e36255161340ec8f1bbf701db29bfc56631d152d302ba74be2303",
    "cross_cats_sorted": [
      "math.AP",
      "math.MP",
      "math.SP"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math-ph",
    "submitted_at": "2025-07-07T13:28:20Z",
    "title_canon_sha256": "167b2048aee550311e0a9f4cf525a798a04b3972aaa4c87de5bb5d8328d668d0"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2507.04988",
    "kind": "arxiv",
    "version": 6
  }
}