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?
