  Numerical approximation of the Boltzmann equation presents a challenging
problem due to its high-dimensional, nonlinear, and nonlocal collision
operator. Among the deterministic methods, the Fourier-Galerkin spectral method
stands out for its relative high accuracy and possibility of being accelerated
by the fast Fourier transform. However, this method requires a domain
truncation which is unphysical since the collision operator is defined in
$\mathbb{R}^d$. In this paper, we introduce a Petrov-Galerkin spectral method
for the Boltzmann equation in the unbounded domain. The basis functions (both
test and trial functions) are carefully chosen mapped Chebyshev functions to
obtain desired convergence and conservation properties. Furthermore, thanks to
the close relationship of the Chebyshev functions and the Fourier cosine
series, we are able to construct a fast algorithm with the help of the
non-uniform fast Fourier transform (NUFFT). We demonstrate the superior
accuracy of the proposed method in comparison to the Fourier spectral method
through a series of 2D and 3D examples.
