Fractal Laplacians on the Unit Interval

We study the eigenvalues and eigenfunctions of the Laplacians on [0, 1] which are defined by bounded continuous positive measures µ supported on [0, 1] and the usual Dirichlet form on [0, 1]. We provide simple proofs of the existence, uniqueness, concavity, and properties of zeros of the eigenfunctions. By rewriting the equation defining the Laplacian as a Volterra-Stieltjes integral equation, we study asymptotic behaviors of the first Neumann and Dirichlet eigenvalues and eigenfunctions as the measure µ varies. For µ defined by a class of post critically finite self-similar structures, we also study asymptotic bounds of the eigenvalues. By restricting µ to a class of singular self-similar measures on [0, 1], we describe both the finite element and the difference approximation methods to approximate numerically the eigenvalues and eigenfunctions. These eigenfunctions can be considered fractal analogs of the classical Fourier sine and cosine functions. We note the existence of a subsequence of rapidly decaying eigenfunctions that are numbered by the Fibonacci numbers.