We study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals which includes 3n-gaskets. We consider such examples as the Sierpinski gasket, a non-p.c.f. analog of the Sierpinski gasket, the level-3 Sierpinski gasket, a fractal 3-tree, the hexagasket, and one dimensional fractals. We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly. We support our theoretical analysis by symbolic and numerical computations.

We show how to calculate the spectrum of the Laplacian operator on fully symmetric, finitely ramified fractals. We consider well-known examples, such as the unit interval and the Sierpiński gasket, and much more complicated ones, such as the hexagasket and a non-post critically finite self-similar fractal. We emphasize the low computational demands of our method. As a conclusion, we give exact formulas for the limiting distribution of eigenvalues (the integrated density of states), which is a purely atomic measure (except in the classical case of the interval), with atoms accumulating to the Julia set of a rational function. This paper is the continuation of the work published by the same authors in Ref. 1.