  For $R>0$, we give a rigorous probabilistic construction on the cylinder
$\mathbb{R} \times (\mathbb{R}/(2\pi R\mathbb{Z}))$ of the (massless)
Sinh-Gordon model. In particular we define the $n$-point correlation functions
of the model and show that these exhibit a scaling relation with respect to
$R$. The construction, which relies on the massless Gaussian Free Field, is
based on the spectral analysis of a quantum operator associated to the model.
Using the theory of Gaussian multiplicative chaos, we prove that this operator
has discrete spectrum and a strictly positive ground state.
