  We analyze a system coupled to a bath of independent harmonic oscillators. We
transform the bath in chain structure by solving an inverse eigenvalue problem.
We solve the equations of motion for the collective variables defined by this
transformation, and we derive the exact dynamics for an harmonic oscillator in
terms of the microscopic motion of the environmental modes. We compare this
approach to the well-known Generalized Langevin Equation and we show that our
dynamics satisfies this equation.
