Pith Number

pith:EJZU6DB4

pith:2026:EJZU6DB467HIGTU5U4GDFH2Q2D

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Two-parameter classes of exactly solvable quantum systems

A. D. Alhaidari

Two-parameter initial values in orthogonal polynomial recursions define new classes of exactly solvable quantum systems whose potentials are realized numerically.

arxiv:2605.09812 v2 · 2026-05-10 · math-ph · math.MP · quant-ph

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Claims

C1strongest claim

We introduce two-parameter classes of exactly-solvable novel systems whose Hamiltonian operators could be represented by tridiagonal symmetric matrices in some orthonormal basis set... a curious phenomenon is observed where bound states and/or resonances are induced in a system with pure continuous spectrum (e.g., a free particle) if the two parameters in the initial values exceed certain critical limits.

C2weakest assumption

That the numerically realized potentials correspond to physically valid, self-adjoint Hamiltonians whose spectra are correctly captured by the orthogonal polynomials generated from the two-parameter recursion, without additional constraints or instabilities arising from the basis choice or truncation.

C3one line summary

Two-parameter families of exactly solvable quantum systems are defined through tridiagonal matrix representations, producing orthogonal polynomials in energy as expansion coefficients, including cases where bound states appear in pure continuous spectra when parameters exceed critical values.

Receipt and verification

First computed	2026-05-28T01:04:42.039877Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

22734f0c3cf7ce834e9da70c329f50d0e86f5673e652684554663611e07df187

Aliases

arxiv: 2605.09812 · arxiv_version: 2605.09812v2 · doi: 10.48550/arxiv.2605.09812 · pith_short_12: EJZU6DB467HI · pith_short_16: EJZU6DB467HIGTU5 · pith_short_8: EJZU6DB4

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/EJZU6DB467HIGTU5U4GDFH2Q2D \
  | 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: 22734f0c3cf7ce834e9da70c329f50d0e86f5673e652684554663611e07df187

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7941239006387db6ba437811de91692d71b543a924565c3e701f246468ede4d1",
    "cross_cats_sorted": [
      "math.MP",
      "quant-ph"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math-ph",
    "submitted_at": "2026-05-10T23:29:07Z",
    "title_canon_sha256": "007327e428ee1cfbbd16681349a457ad4db502ce6dd45f0a122c5fa4de3b0994"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.09812",
    "kind": "arxiv",
    "version": 2
  }
}