Eigenfunction completeness
QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_completeness
Project documentation
Assuming Plancherel's theorem (which is not yet in Mathlib), the topological closure of the span of the eigenfunctions of the harmonic oscillator is the whole Hilbert space. The proof of this result relies on fourierIntegral_zero_of_mem_orthogonal and Plancherel's theorem which together give us that the norm of f x * e ^ (- x^2 / (2 * ξ^2)) is zero fo...
Exact Lean statement
theorem eigenfunction_completeness
(plancherel_theorem : ∀ {f : ℝ → ℂ} (hf : Integrable f volume) (_ : MemLp f 2),
eLpNorm (𝓕 f) 2 volume = eLpNorm f 2 volume) :
(Submodule.span ℂ
(Set.range (fun n => HilbertSpace.mk (Q.eigenfunction_memHS n)))).topologicalClosure = ⊤Formal artifact
Lean source
theorem eigenfunction_completeness (plancherel_theorem : ∀ {f : ℝ → ℂ} (hf : Integrable f volume) (_ : MemLp f 2), eLpNorm (𝓕 f) 2 volume = eLpNorm f 2 volume) : (Submodule.span ℂ (Set.range (fun n => HilbertSpace.mk (Q.eigenfunction_memHS n)))).topologicalClosure = ⊤ := by rw [Submodule.topologicalClosure_eq_top_iff] refine (Submodule.eq_bot_iff (Submodule.span ℂ (Set.range (fun n => HilbertSpace.mk (Q.eigenfunction_memHS n))))ᗮ).mpr ?_ intro f hf apply Q.zero_of_orthogonal_eigenVector f ?_ plancherel_theorem intro n rw [@Submodule.mem_orthogonal'] at hf rw [← inner_conj_symm] have hl : ⟪f, HilbertSpace.mk (Q.eigenfunction_memHS n)⟫_ℂ = 0 := by apply hf refine Finsupp.mem_span_range_iff_exists_finsupp.mpr ?_ use Finsupp.single n 1 simp rw [hl] simp- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/Completeness.lean:447-466
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