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 filling the empty sections C.1 (kinetic energy), C.2 (potential energy), C.3 (Hamiltonian) and D (as a quantum system) of that file on the model of SpaceDQuantumSystem, 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 C (operators). kineticCLM = (2m)⁻¹ 𝐩 ⬝ᵥ 𝐩 with kineticCLM_apply (dimension one), potentialCLM = smulLeftCLM ℂ (ofReal ∘ potential) with potentialCLM_apply, hamiltonianCLM with hamiltonianCLM_apply; the unbounded kineticOperator, potentialOperator (with potentialOperator_isSelfAdjoint and schwartzSubmodule_le_potentialOperator_domain) and hamiltonianOperator. 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).

Section D. toSpaceDQuantumSystem : SpaceDQuantumSystem of dimension 1, with toSpaceDQuantumSystem_hamiltonianCLM and toSpaceDQuantumSystem_hamiltonianOperator (both rfl), so that the generic results on SpaceDQuantumSystem apply to this system.

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 and the table of contents are filled in.

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.

🤖 Generated with Claude Code

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