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The most realistic situations in quantum mechanics involve the interactionbetween two or more systems. In the most of reliable models, the form andstructure of the interactions generate differential equations which are, in themost of cases, almost impossible to solve exactly. In this paper, using theSchwinger Quantum Action Principle, we found the time transformation functionthat solves exactly the harmonic oscillator interacting with a set of otherharmonic coupled oscillators. In order to do it, we have introduced a newspecial set of creation and annihilation operators which leads directly to the\emph{dressed states} associated to the system, which are the real quantumstates of the interacting \emph{\textquotedblleftfield-particle\textquotedblright} system. To obtain the closed solution, it isintroduced in the same foot a set of \emph{normal mode} creation andannihilation operators of the system related to the first ones by an orthogonaltransformation. We find the eigenstates, amplitude transitions and the systemspectra without any approximation. At last, we show that the SchwingerVariational Principle provides the solutions in a free representation way.

Non-perturbative Solution to the Quantum Interaction Problem via Schwinger's Action Principle