# Conditions on the Hamiltonian of a classical system that yeild essentially self-adjoint quantum Hamiltonian

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 $$H\to\mathcal{H}$$.