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Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve?

In particular, if $0$ is in the spectrum, does this tell us anything about the solvability of the Dirichlet or Poisson problem for the region? This second question is motivated by the following theorem that I found while flipping through Davies' Spectral Theory and Differential Operators. From the emphasis he puts on it and the amount of work he did to prove it, I feel it must be important, but I can't say why.

Let $\Omega\subset \mathbb R^2$ be regular, and let $H$ be the Friedrichs extension of $\small -\triangle$ initially defined on $C^\infty_c(\Omega)$. Then $0\in \operatorname{Spec} H$ if and only if the inradius of $\Omega$ is infinite.

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Are you asking what the spectrum of Laplacian on $\Omega$ tells us about $\Omega$ itself ? –  nonlinearism Jan 25 at 0:11
    
@nonlinearism Not really. I know that's a big, famous problem. I'm looking for more basic things, especially relating to the Dirichlet and Poisson PDEs. –  Potato Jan 25 at 0:26
    
$+1$ I've been wondering this for a long while. The eigenvalues of the Laplacian are apparently important in Riemannian geometry, but like you, I have no idea why. –  Jesse Madnick Jan 25 at 5:37
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Related: math.stackexchange.com/questions/454922/… –  Tomás Jan 25 at 11:18

1 Answer 1

The spectrum of $-\Delta$ reveals properties of light, heat, sound, and atomic phenomena. I'll briefly summarize.

The heat equation $$ \frac{\partial u}{\partial t} = c\Delta u, $$ can be solved if you have a complete basis of eigenfunctions for $\Delta u$. The eigenvalues give the time decay rates of the eigenfunctions in time. More explicitly, if $(-\Delta) e_{\lambda} = \lambda e_{\lambda}$, then $$ u(x,t)=e^{-c\lambda t}e_{\lambda}(x) $$ is a solution of the heat equation, and, in principle, a full solution can be built from linear (discrete or continuous) sums of such solutions. This comes from the classical problem of separation of variables originally proposed by Fourier in his Treatise on Heat Conduction. Spectral Theory has its origins here.

In a similar way, one may develop solutions of the wave equation: $$ \frac{\partial^{2} u}{\partial t^{2}}=\frac{1}{c^{2}}\Delta u. $$ The wave equation governs the time-dependent behavior of the electric field in a homogenous, isotropic, source-free material. (The propogation of light is described through the wave equation.) In this case the eigenvalues determine the modulation frequency for a particular eigenfunction through separation of variables. This corresponds to classical notions of prism and spectrum. For a string fixed at two endpoints and undergoing small vibrations, the classical one-dimensional wave equation describes mechanical displacements, and gives a fundamental vibrational mode and all integer multiples of those modes (a.k.a. harmonics.) This relates to sound, and it's almost certain that this is the origin of the term "harmonic anaysis." The harmonic frequencies are derived from the eigenvalues of $\frac{d^{2}}{dx^{2}}$. A circular drum head is similarly analyzed from the eigenvalues and eigenvectors of the 2-d Laplacian. The resonant frequencies of the drum are non-harmonic and are derived from the eigenvalues of the Laplacian.

The non-relativistic hydrogen atom has a corresponding Quantum Mechanical Schrodinger Wave equation $$ i\hbar\frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2\mu}\Delta \psi + \frac{e^{2}}{4\pi \epsilon_{0}r}\psi . $$ Though the problem is not fully reduced to eigenvalues/eigenvectors of $\Delta$, the eigenvalues of $\Delta$ on functions defined on the unit spherical shell (spherical harmonics) have to do with the permissible values of the orbital energies of the atom; this reveals much about the orbital shells, and the spectral lines of the Hydrogen atom. Remember S, P, D, F from Chemistry? http://en.wikipedia.org/wiki/Atomic_orbital.

There are applications to fluids as well.

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Thanks for the information. Could you say why it is important to know if $0$ is in the spectrum of the Laplacian or not? Does it have anything to do with the solvability of the Dirichlet or Poisson problem on a region? –  Potato Jan 25 at 6:42
    
I suppose that if we know it isn't, we can always invert it and solve the equation. At least, this seems to get a "weak" solution. Right? –  Potato Jan 25 at 7:14
    
@Potato: $(-\Delta f,f) \ge 0$ for $f \in C^{\infty}_{c}(\Omega)$. The Friedrichs extension of $-\Delta$ is guaranteed to have the same lower bound as the original operator: i.e., if $(-\Delta f,f) \ge \epsilon(f,f)$ for some $\epsilon$ and all $f$, this bound is preserved by the Friedrichs extension. $\epsilon=0$ on free space. Preserving this bound was the original goal of the Friedrichs extension and answered an open question of this type at the time. That's why they mention that bound. Yes, it has to do with the invertibility of $\Delta_{f}$ and the continuity of the inverse. –  T.A.E. Jan 25 at 12:51
    
Could you explain the invertibility issue a little more? For example, if I want to solve $\small \Delta f=0$ in a disk, it seems like naively inverting just gives $f=\Delta^{-1}0=0$. So the boundary conditions have to come into play somehow. –  Potato Jan 25 at 19:31
    
To completely define the eigenvalue problem or to find the solution of Laplace PDE, you have define the domain in which the solutions lie. For example, you need to specify the boundary value of Dirichlet solution, and boundary flux for Neumann. The solution and the spectrum is different for Dirichlet problem, compared to Neumann problem. Also required are the smoothness properties of the desired solutions. –  nonlinearism Jan 25 at 19:47

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