Characterization of Other Quantum Limits on the Sierpinski Gasket

Characterize the other measures that can occur as weak limits of probability measures associated with sequences of high-energy eigenfunctions of the standard Laplacian on the Sierpinski gasket.

Background

The paper proves that, for every point in a dense subset of the Sierpinski gasket, there exists a sequence of high-energy eigenfunctions of the standard Laplacian whose associated probability measures converge weakly to the Dirac measure at that point. This demonstrates that singular measures can arise as quantum limits on the Sierpinski gasket, unlike in compact Riemannian manifolds.

The authors leave unresolved the broader classification problem: beyond the Dirac measures constructed in the paper, it is unknown which other measures may arise as weak limits of probability measures associated with high-energy Laplacian eigenfunctions on the Sierpinski gasket. Such a characterization would describe the possible quantum-limit measures for this fractal setting.