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We present exact solutions for some eigenstates of hopping models on one andtwo dimensional quasiperiodic tilings and show that they are "critical" states,by explicitly computing their multifractal spectra. These eigenstates are shownto be generically present in 1D quasiperiodic chains, of which the Fibonaccichain is a special case. We then describe properties of the ground states for aclass of tight-binding Hamiltonians on the 2D Penrose and Ammann-Beenkertilings. Exact and numerical solutions are seen to be in good agreement.

Critical eigenstates and their properties in one and two dimensional quasicrystals