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I'm reading about the notion of spectral dimension which is a measure of how particles diffuse in some space at different scales. An important aspect of spectral dimension is the eigenvalues/eigenfunctions of the Laplacian operator which in some sense determine what scale the diffusion process is probing. I was wondering if there was some geometrical interpretation of the eigenfunctions of the Laplacian that can give some how they relate to the notion of scale.

Pages 3-5 of this paper explain the context of my question.

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  • $\begingroup$ This seems really unclear and vague. The eigenfunctions of the Laplace operator on $\Bbb R$ (with the usual metric) are complex exponentials. I'm not sure how these can encode anything about scale. $\endgroup$ – Cameron Williams Jul 3 '15 at 21:07
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    $\begingroup$ arxiv.org/pdf/0911.0401v2.pdf I guess I'm vague because I'm confused why the eigenvalues/eigenfunctions are even used in the discussion in the first place. Read pages 3-5 (the rest of the paper is physics stuff). It initially discuss the notion of spectral dimension on smooth manifolds as motivation for the adapted case on simplicial manifolds. $\endgroup$ – SWV Jul 3 '15 at 21:57
  • $\begingroup$ Where are you considering the Laplace operator? In a bounded domain of a Euclidean space, a full Euclidean space, or something else? $\endgroup$ – Joonas Ilmavirta Jul 3 '15 at 22:03
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    $\begingroup$ In the paper I linked in the previous comment, it says that we are interested in the Laplacian operator on d-dimensional closed Riemannian manifolds. $\endgroup$ – SWV Jul 3 '15 at 22:08
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There are two scales at play here. Let $M$ be a closed Riemannian manifold and let $\Delta$ be the Laplace-Beltrami operator on it. There is an orthonormal eigenbasis $\{\phi_k\}_{k=1}^\infty$ of $L^2(M)$ and an increasing sequence of eigenvalues $\lambda_k\geq0$ so that $\Delta\phi_k=-\lambda_k\phi_k$.

The piece of an article you linked to discusses the heat equation (and the related heat trace). For any $k$, the function $u_k(x,t)=e^{-\lambda_kt}\phi_k(x)$ solves the heat equation $\partial_tu=\Delta u$. (You can expand a more general initial condition in terms of the eigenfunctions and write a solution to the heat equation as a series of $u_k$s.) The eigenvalue $\lambda_k$ determines the time scale of the decay of the eigenfunction $\phi_k$ under the heat equation; if $\lambda_k$ is large, then $u_k\to0$ quickly as $t\to\infty$.

On the other hand, $\lambda_k$ is related to the scale of oscillations of $\phi_k$. This is most transparent in the one dimensional case $M=\mathbb R/2\pi\mathbb Z$ (the unit circle). The eigenfunctions are $\phi_m(x)=e^{imx}$ with eigenvalue $\lambda_m=m^2$. (Here $m$ ranges over $\mathbb Z$ for convenience. Other eigenvalues than zero are degenerate.) The number $m^2$ describes how rapidly the function $\phi_m$ oscillates, right? The picture is similar in any dimension, but it is generally difficult to find explicit eigenfunctions.

That is, short time scales correspond to long distance scales (and both correspond to low eigenvalues) and vice versa. Does this interplay between the two scales and their relation to the Laplacian answer your question?

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    $\begingroup$ Are the eigenfunctions oscillitory in some uniform way so that it "represents" the same scale throughout the manifold? $\endgroup$ – SWV Jul 3 '15 at 23:27
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    $\begingroup$ @SWV, the second derivative describes the size of oscillations, so in that sense, yes. But in higher dimensions the oscillations oscillations can take place in different directions. For example on the flat torus $\mathbb R^2/2\pi\mathbb Z^2$ the functions $\phi(x)=e^{5ix_1}$ and $\psi(x)=e^{3ix_1+4ix_2}$ have the same eigenvalue 25. In some sense they oscillate the same amount. But still they represent the same scale throughout the manifold, and this happens on all manifolds (although the "length scale of oscillation" is not a perfectly well-defined thing). $\endgroup$ – Joonas Ilmavirta Jul 4 '15 at 8:12

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