Harmonic Oscillators on Infinite Sierpinski Gaskets

Abstract

We study the analog of the quantum-mechanical harmonic oscillator on infinite blowups of the Sierpinski Gasket, using the standard Kigami Laplacian. Our main task is to find a class of potentials analogous to ${\frac{1}{2}(x-x_0)^2}$ on the line. We describe a class of potentials u with the properties Δu = 1, u attains a minimum value zero, and u → ∞ at infinity. We show how to construct such potentials attaining the minimum value at any prescribed point, and we show how to parameterize the class of potentials by a certain surface in ${\mathbb{R}^3}$. We obtain estimates for the growth rate of the eigenvalue counting function for  −Δ + u. We obtain numerical approximations to the eigenfunctions, and in particular observe that the ground-state eigenfunction resembles a Gaussian function.