Spectral problem for a harmonic chain with dissipation at the boundaries

We consider a spectral problem for a dynamical system describing, in the Schrödinger variables, the motion in a finite homogeneous chain of coupled harmonic oscillators with boundary conditions that admit dissipation and energy pumping in the system. The solution of this problem is given for arbitrary parameter values in the boundary conditions and for any sufficiently large number of oscillators in the chain.