Refinement trajectory and determination of eigenstates by a wavelet based adaptive method. 
The detail structure of the wave function is analyzed at various refinement levels using the methods of wavelet analysis.  The eigenvalue problem of a model system is solved in granular Hilbert spaces, and the trajectory of the eigenstates is traced in terms of the resolution.  An adaptive method is developed for identifying the fine structure localization regions, where further refinement of the wave function is necessary.