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### Open Access Journals

CPAA

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.

CPAA

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.

CPAA

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.

CPAA

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.

CPAA

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 $\mathcal{VS}_n$.)

CPAA

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.

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