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 $H^\dagger$. In the analysis carried out here, the role of {\em weak pseudo-bosonic} ladder operators is relevant.