7.5 The Hermite–Sobolev chain
The harmonic oscillator \(A f = -f'' + (x^2/4 + 1/2)\, f\) is a continuous linear map \(\mathcal{S}(\mathbb {R},\mathbb {R}) \to \mathcal{S}(\mathbb {R},\mathbb {R})\), built from Mathlib’s Laplacian CLM and multiplication by the temperate-growth multiplier \(x^2/4 + 1/2\). It is symmetric for the \(L^2\) pairing, \(\int (Af)\, g = \int f\, (Ag)\), and the Hermite functions are its eigenfunctions: \(A h_n = (n+1)\, h_n\), proved from the ladder identities for \(h_n\).
The \(n\)-th Hermite coefficient \(\varphi \mapsto \int h_n \varphi \) is a continuous linear functional on \(\mathcal{S}(\mathbb {R},\mathbb {R})\), and the level-\(r\) Hermite–Sobolev seminorm is \(\| \varphi \| _r = \bigl(\sum _n (n+1)^{2r} \langle h_n, \varphi \rangle ^2\bigr)^{1/2}\); the defining series is summable because coefficient decay \(|\langle h_n, \varphi \rangle | \lesssim (n+1)^{-r}\) at every order follows from powers of the oscillator via the eigenrelation and self-adjointness. The same quantity is packaged as a bundled Seminorm by factoring through the weighted coefficient map into \(\ell ^2\).
Each Hermite–Sobolev seminorm is continuous for the canonical Schwartz topology: for every \(r\) there are \(C \ge 0\) and a finite set \(s\) of indices with \(\| \varphi \| _r \le C \cdot \sup _{(k,n) \in s} p_{k,n}(\varphi )\). Conversely, every canonical Schwartz seminorm is dominated by a single Hermite–Sobolev level: \(p_{k,m}(\varphi ) \le C\, \| \varphi \| _r\) for some \(C \ge 0\) and \(r\) (obtained by trading powers of \(x\) and derivatives against oscillator powers). Hence the countable Hilbertian chain \((\| \cdot \| _r)_{r \in \mathbb {N}}\) generates the Schwartz topology.
The vectors \(e^r_j = (j+1)^{-r}\, h_j\) form the \(\| \cdot \| _r\)-orthonormal system, and \(\| e^r_j\| _q = (j+1)^{q-r}\). Whenever \(q + 1 \le r\), the squares are summable: \(\sum _j \| e^r_j\| _q^2 = \sum _j (j+1)^{2(q-r)} {\lt} \infty \). This is the nuclearity input (the inclusion between consecutive levels of the chain is Hilbert–Schmidt) consumed by the Gaussian-averaging step of Mitoma’s argument.