  The long-time asymptotics is analyzed for all finite energy solutions to a
model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the
nonlinearity concentrated at a single point: each finite energy solution
converges as time goes to plus or minus infinity to the set of all ``nonlinear
eigenfunctions'' of the form $\psi(x)e\sp{-i\omega t}$. The global attraction
is caused by the nonlinear energy transfer from lower harmonics to the
continuous spectrum and subsequent dispersive radiation.
  We justify this mechanism by the following novel strategy based on inflation
of spectrum by the nonlinearity. We show that any omega-limit trajectory has
the time-spectrum in the spectral gap [-m,m] and satisfies the original
equation. This equation implies the key spectral inclusion for spectrum of the
nonlinear term. Then the application of the Titchmarsh Convolution Theorem
reduces the spectrum of each omega-limit trajectory to a single harmonic in
[-m,m].
  The research is inspired by Bohr's postulate on quantum transitions and
Schroedinger's identification of the quantum stationary states to the nonlinear
eigenfunctions of the coupled U(1)-invariant Maxwell-Schroedinger and
Maxwell-Dirac equations.
