Average error for spectral asymptotics on surfaces
Robert S. Strichartz
Let $N(t)$ denote the eigenvalue counting function of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula $\widetilde N(t)=At+Bt^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t)=\frac 1 t \int^t_0 D(s)\,ds$ for $D(t)=N(t)-\widetilde N(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture.
keywords: Spectral asymptotics almost periodic functions. Laplacian cone point singularities surfaces of constant curvature average error
Average number of lattice points in a disk
Sujay Jayakar Robert S. Strichartz
The difference between the number of lattice points in a disk of radius $\sqrt{t}/2\pi$ and the area of the disk $t/4\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \leq t \leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.
keywords: Bessel function. Lattice points Weyl asymptotics
Hodge-de Rham theory on fractal graphs and fractals
S. Aaron Z. Conn Robert S. Strichartz H. Yu
We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and 3-dimensional Sierpinski gasket (SG$^3$), but the method is expected to be effective for many PCF fractals, and also infinitely ramified fractals such as the Sierpinski carpet (SC). Our approach is to construct k-forms and de Rham differential operators $d$ and $\delta$ for a sequence of graphs approximating the fractal, and then pass to the limit with suitable renormalization, in imitation of Kigami's approach on constructing Laplacians on functions. One of our results is that our Laplacian on 0-forms is equal to Kigami's Laplacian on functions. We give explicit construction of harmonic 1-forms for our examples. We also prove that the measures on line segments provided by 1-forms are not absolutely continuous with respect to Lebesgue measures.
keywords: Analysis on fractals Sierpinski gasket k-forms harmonic 1-forms Hodge-de Rham theory fractal graphs.
A fractal quantum mechanical model with Coulomb potential
Robert S. Strichartz
We study the Schrödinger operator $ H = - \Delta + V $ on the product of two copies of an infinite blowup of the Sierpinski gasket, where $ V$ is the analog of a Coulomb potential ($\Delta V$ is a multiple of a delta function). So $H$ is the analog of the standard Hydrogen atom model in nonrelativistic quantum mechanics. Like the classical model, we show that the essential spectrum of $H$ is the same as for $ - \Delta $, and there is a countable discrete spectrum of negative eigenvalues.
keywords: Schrödinger operator with Coulomb potential nonrelativistic Hydrogen atom model Analysis on fractals Sierpinski gasket
Laplacians on a family of quadratic Julia sets II
Tarik Aougab Stella Chuyue Dong Robert S. Strichartz
This paper continues the work started in [4] to construct $P$-invariant Laplacians on the Julia sets of $P(z) = z^2 + c$ for $c$ in the interior of the Mandelbrot set, and to study the spectra of these Laplacians numerically. We are able to deal with a larger class of Julia sets and give a systematic method that reduces the construction of a $P$-invariant energy to the solution of nonlinear finite dimensional eigenvalue problem. We give the complete details for three examples, a dendrite, the airplane, and the Basilica-in-Rabbit. We also study the spectra of Laplacians on covering spaces and infinite blowups of the Julia sets. In particular, for a generic infinite blowups there is pure point spectrum, while for periodic covering spaces the spectrum is a mixture of discrete and continuous parts.
keywords: Laplacians. Julia sets
Analysis of the Laplacian and spectral operators on the Vicsek set
Sarah Constantin Robert S. Strichartz Miles Wheeler
We study the spectral decomposition of the Laplacian on a family of fractals $\mathcal{VS}_n$ that includes the Vicsek set for $n=2$, extending earlier research on the Sierpinski Gasket. We implement an algorithm [23] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann boundary conditions. We study gaps in the ratios of eigenvalues and eigenvalue clusters. We give an explicit formula for the Green's function on $\mathcal{VS}_n$. Finally, we explain how the spectrum of the Laplacian on $\mathcal{VS}_n$ converges as $n \to \infty$ to the spectrum of the Laplacian on two crossed lines (the limit of the sets $\mathcal{VS}_n$.)
keywords: Vicsek set Laplacians on fractals Green's function eigenvalue ratio gaps. eigenvalue clusters wave propagators Weyl ratio spectral operators heat kernels

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