Spectral Selection via Dissipative Normalized Flows

We analyze Rayleigh-type normalized dissipative flows associated with nonnegative self-adjoint operators on a Hilbert space. The Rayleigh quotient is shown to be a Lyapunov functional whose dissipation is entirely governed by a residual measuring deviation from the spectral set. This identity yields a rigidity principle: vanishing dissipation forces the state to be spectral, and every ω–limit point is an eigenstate under mild compactness assumptions. We further show that spectral structure promotes rigidity to selection. In finite dimension, the flow converges to the normalized projection onto the first eigenspace present in the initial spectral support, with exponential rates controlled by spectral gaps. An analogous selection result is proved in infinite dimension under compact resolvent, covering a broad class of PDE generators. Explicit toy models illustrate the resulting coarse-to-fine relaxation mechanism