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What are the conditions on the Hamiltonian of a classical system that under these conditions the quantum Hamiltonian obtained via Weyl quantization will be essentially self-adjoint in $L_2(\mathbb{R}^d)$ on the domain consisting of all infinitely smooth compactly supported functions? $$ (\mathcal{H}\psi)(x)=\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}e^{2\pi i(x-q)p}H\left(\frac{x+q}{2},p\right)\psi(q)dq\,dp $$

I'm looking for a reference book or a paper to cite. I'm also interested in properties of a linear map $H\to\mathcal{H}$.

Thank you in advance!

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  • $\begingroup$ Have you looked in G. Folland's book "Harmonic Analysis in Phase Space"? $\endgroup$
    – Matt Rosenzweig
    Commented Aug 13, 2019 at 17:08
  • $\begingroup$ @MattRosenzweig, thanks for the suggestion, I will look into it. Could you please specify the number of page/theorem if possible $\endgroup$
    – Glinka
    Commented Aug 14, 2019 at 13:44
  • $\begingroup$ @MattRosenzweig, Your reference is very useful. It has what I'm looking for, but I have a feeling that conditions there can be weakened. Also, the way they are stated is unclear: it says that the function's growth should be bounded by a polynomial, but nothing about smoothness or even measurability, though clearly for non-measurable functions nothing holds. $\endgroup$
    – Glinka
    Commented Aug 25, 2019 at 8:08
  • $\begingroup$ @MattRosenzweig I think conditions can be weakened by considering the space of bump functions (the case I'm interested in) instead of Schwartz space $\endgroup$
    – Glinka
    Commented Aug 25, 2019 at 8:09

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