feat(QuantumMechanics): Hamiltonian and related operators of the Pöschl-Teller system

[KevorkianPhilippe]

Resolves the TODO "Define the Hamiltonian and related operators for the Pöschl-Teller quantum system" of QuantumMechanics/PoschlTeller/Basic.lean, by realising the system as a SpaceDQuantumSystem, whose kinetic, potential and Hamiltonian operators are then the operators of the system, and by adding the properties of the potential that make these operators usable.

Section A (potential). New depth = ℏ² κ² N (N + 1) / (2 m) with potential_eq (V = -depth / cosh² (κ x)), potential_eq_tanh (V = -depth (1 - tanh² (κ x))), depth_pos, potential_neg, neg_depth_le_potential, potential_zero, potential_neg_eq (even), and the regularity lemmas tanh_hasTemperateGrowth, ofReal_comp_tanh_hasTemperateGrowth, potential_hasTemperateGrowth (from tanh_const_mul_hasTemperateGrowth composed with the coordinate), ofReal_comp_potential_hasTemperateGrowth, potential_continuous.

Section B (as a quantum system). toSpaceDQuantumSystem : SpaceDQuantumSystem of dimension 1 (an abbrev), with toSpaceDQuantumSystem_d, toSpaceDQuantumSystem_m and toSpaceDQuantumSystem_potential (all rfl). The kinetic, potential and Hamiltonian operators (Schwartz and unbounded versions) are those of SpaceDQuantumSystem; they are not redefined here.

Section C (operators). For the operators of Q.toSpaceDQuantumSystem: toSpaceDQuantumSystem_kineticCLM_apply (dimension one), toSpaceDQuantumSystem_potentialCLM_apply, toSpaceDQuantumSystem_potentialOperator_isSelfAdjoint and toSpaceDQuantumSystem_potentialOperator_domain_ge (the generic lemmas, with their hypotheses discharged by section A), toSpaceDQuantumSystem_hamiltonianCLM_apply. For the existing ladder operators: tanhCLM_apply, creationCLM_apply, annihilationCLM_apply, and the relations creationCLM_add_annihilationCLM (= (2/√(2m)) 𝐩 0) and creationCLM_sub_annihilationCLM (= (2 i ℏ κ/√(2m)) tanhCLM).

Not touched: the eigensystem via the ladder operators and the reflectionless property (the two other TODOs), which need the commutator of 𝐩 with tanh (κ X).

Changes to existing content: only the TODO line removed, the SpaceDQuantumSystem import added, and MeasureTheory, SpaceDHilbertSpace opened; the key results are filled in, and the table of contents is reorganised (the empty sections C.1-C.3 and D became sections B and C.1).

Checks: lake build, lint-style.sh, runPhyslibLinters, check_file_imports, codespell; standard axioms only.

AI disclosure. The statements were frozen first (with sorry) and their proofs were produced by Kimi K2.7 Code (Moonshot AI) running in the DeepSeek Harness against physlib, then rewritten and shortened by Claude Fable 5.1 (Claude Code) for the physlib style (in particular the continuity of the potential now follows from its temperate growth); a type-identity check between the frozen statements and the final lemmas was compiled. I reviewed the statements, the docstrings and the proofs. The refactor of Sept 24 (operators taken from toSpaceDQuantumSystem) was made with Claude Opus 5.5 (Claude Code).

Update (Sept 24). Following the review, the kinetic, potential and Hamiltonian operators are no longer defined a second time: they are those of Q.toSpaceDQuantumSystem, and the file keeps only the facts specific to this system. Master is merged in.

🤖 Generated with Claude Code

[github-actions]

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[nateabr]

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[github-actions]

Review claim by @nateabr completed — thanks for the review.

[github-actions]

@nateabr about 24 hours are left on your review claim for this PR, which runs out at 2026-09-23 07:28 UTC.

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[github-actions]

@nateabr your review claim on this PR ran out at 2026-09-23 07:28 UTC without a review, so I have removed you as a reviewer and assignee.

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[nateabr]

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[nateabr]

This PR looks great, I have added one question as a comment

[nateabr]

What is the intended benefit of maintaining this parallel API (including potentialCLM/hamiltonianCLM)? Since these defs are definitionally equal to the corresponding toSpaceDQuantumSystem operators, could we derive them directly from Q.toSpaceDQuantumSystem instead?

[nateabr]

awaiting-author

[KevorkianPhilippe]

Thanks. There was no benefit: the file had empty sections for these operators, and the bridge to
SpaceDQuantumSystem was only added afterwards. They are now taken from it (6bb6eff):
toSpaceDQuantumSystem is an abbrev placed right after the potential, the six definitions are
removed, and the file keeps only the facts specific to this system, stated for
Q.toSpaceDQuantumSystem (pointwise formulas in dimension one, self-adjointness and domain of the
potential operator). Master is also merged in, and the description is updated.

Drafted with Claude, reviewed and validated by me.

-awaiting-author

[nateabr]

Changes look good, thanks. Please rebase onto the latest version of physlib.

[jstoobysmith]

@KevorkianPhilippe Would you mind rebase onto the latest version of physlib before we merge this (see @nateabr's comment above).

awaiting-author

[KevorkianPhilippe]

Done: master is merged in (ba6fc10), without conflict. I merged rather than rebased to avoid a force-push; happy to rebase if you prefer.

Drafted with Claude, reviewed and validated by me.

-awaiting-author