  We introduce a novel virtual element method (VEM) for the two dimensional
Helmholtz problem endowed with impedance boundary conditions. Local
approximation spaces consist of Trefftz functions, i.e., functions belonging to
the kernel of the Helmholtz operator. The global trial and test spaces are not
fully discontinuous, but rather interelement continuity is imposed in a
nonconforming fashion. Although their functions are only implicitly defined, as
typical of the VEM framework, they contain discontinuous subspaces made of
functions known in closed form and with good approximation properties (plane
waves, in our case). We carry out an abstract error analysis of the method, and
derive $h$-version error estimates. Moreover, we initiate its numerical
investigation by presenting a first test, which demonstrates the theoretical
convergence rates.
