  Methanol is one of the most abundant interstellar Complex Organic Molecules
(iCOMs) and it represents a major building block for the synthesis of
increasingly complex oxygen-containing molecules. The reaction between
protonated methanol and its neutral counterpart, giving protonated dimethyl
ether, (CH$_3$)$_2$OH$^+$, along with the ejection of a water molecule, has
been proposed as a key reaction in the synthesis of dimethyl ether in space.
Here, gas phase vibrational spectra of the (CH$_3$)$_2$OH$^+$ reaction product
and of the [C$_2$H$_9$O$_2$]$^+$ intermediate complex(es), formed under
different pressure and temperature conditions, are presented. The widely
tunable free electron laser for infrared experiments, FELIX, was employed to
record their vibrational fingerprint spectra using different types of infrared
action spectroscopy in the $600-1700$ cm$^{-1}$ frequency range, complemented
with measurements using an OPO/OPA system to cover the O-H stretching region
$3400-3700$ cm$^{-1}$. The formation of protonated dimethyl ether as a product
of the reaction is spectroscopically confirmed, providing the first gas-phase
vibrational spectrum of this potentially relevant astrochemical ion.
