What's the relation between frequency band of $X(j\omega)$ and $\Phi_{xx}(j\omega)$?
The autocorrelation of $x(t)$ is
$$r_x(t)=x(t)\star x(-t)\tag{1}$$
where $\star$ denotes convolution. Taking the Fourier transform of $(1)$ gives
$$S_x(\omega)=X(\omega)X^*(\omega)=|X(\omega)|^2\tag{2}$$
$S_x(\omega)$ is the energy density of $x(t)$, and according to $(2)$ it equals the squared magnitude of the Fourier transform of $x(t)$. So if $x(t)$ is band-limited, both $X(\omega)$ and $S_x(\omega)$ are zero outside the signal's bandwidth.
Note that a deterministic continuous signal which has a Fourier transform (represented by an ordinary function) is usually an energy signal, which doesn't have a power spectrum (only an energy density).