  We develop spectral theory for the generator of the q-Boson (stochastic)
particle system. Our central result is a Plancherel type isomorphism theorem
for this system. This theorem has various implications. It proves the
completeness of the Bethe ansatz for the q-Boson generator and consequently
enables us to solve the Kolmogorov forward and backward equations for general
initial data. Owing to a Markov duality with q-TASEP, this leads to moment
formulas which characterize the fixed time distribution of q-TASEP started from
general initial conditions. The theorem also implies the biorthogonality of the
left and right eigenfunctions.
  We consider limits of our q-Boson results to a discrete delta Bose gas
considered previously by van Diejen, as well as to another discrete delta Bose
gas that describes the evolution of moments of the semi-discrete stochastic
heat equation (or equivalently, the O'Connell-Yor semi-discrete directed
polymer partition function). A further limit takes us to the delta Bose gas
which arises in studying moments of the stochastic heat equation /
Kardar-Parisi-Zhang equation.
