Multiplication of distributions in a linear gain and loss system

We consider a model of coupled oscillators which can be seen as a gain and loss system. In the attempt to quantize the system, we propose a new definition of multiplication between distributions, and we check that this definition can be adopted when checking the biorthonormality of the eigenstates of the Hamiltonian H of the system, and of its adjoint H-& DAG;. In the analysis carried out here, the role of weak pseudo-bosonic ladder operators is relevant.