# Equivalence of Hilbert space norm associated to the harmonic oscillator and a sum of Sobolev and weighted $L^2$ norms

I have seen an equivalence claimed in a few places, but I do not know of a reference that actually proves it with details and it has been a while since I took graduate courses on all this. Apologies if this is something standard. A reference would be appreciated, particularly if it illuminates the general picture.

The simplest concrete version of my question is the following. Let

$$\mathcal{H} = -\Delta + |x|^2$$

be the quantum harmonic oscillator Hamiltonian in $$\mathbb{R}^d$$ and for $$k\in\mathbb{N}$$ define a Hilbert space by taking the closure of the Schwarz class functions $$\mathcal{S}(\mathbb{R}^d)$$ in the norm defined by $$\|f\|_k^2 = \int_{\mathbb{R}^d}|\mathcal{H}^k f(x)|^2dx$$ I have seen it claimed in a few places (for example Proposition 2.3 in https://aip.scitation.org/doi/abs/10.1063/1.5048726?journalCode=jmp) that this norm is equivalent to the norm given by a combination of a weighted $$L^2$$ norm and an ordinary Bessel-Sobolev norm $$|||f|||_k^2 = \|\mathcal{F}^{-1}(1+|\xi|^2)^{k}\mathcal{F}f\|^2_{L^2(\mathbb{R}^d)} + \| |x|^{2k}f \|^2_{L^2(\mathbb{R}^d)}$$ I think I see that $$\|\cdot\|_k$$ is controlled by $$|||\cdot|||_k$$, but I do not know how to prove the reverse inequality.

Based on how I have seen this talked about (for example, the proof of that proposition in that reference) my understanding is that there is a simple proof of this that begins by viewing $$\mathcal{H}$$ as the operator with Fourier symbol given by $$a(x,\xi) = 4\pi^2|\xi|^2+|x|^2$$.

Does anyone here know how such a proof would go? Or of any proof of this claim?