Research proof index

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...

The eigenfunctions are integrable.

A simplification of the product of two eigenfunctions.

The eigenfunction are normalized.

The eigenfunctions of the quantum harmonic oscillator are orthogonal.

The eigenfunctions are square integrable.