What is the purpose of computing the eigenvalue of a PDE problem?

I understand that eigenvalues have their purpose in linear algebra (e.g. iterative methods won't converge unless the modulus of the spectral radius is less than or equal to one). But when I solve partial differential equations using a finite difference scheme, I'm generally more interested in the solution, its stability, and its convergence. What insight can I gain from solving an eigenvalue problem from the same PDE?

Say, for example, the eigenvalue problem

$\nabla^2u=\lambda u$ in $\Omega$
$u=0$ on $\partial \Omega$

Without a context to a physical problem, does solving this eigenvalue problem provide any insight into other related problems such as

$\nabla^2u=0$ in $\Omega$
$u=0$ on $\partial \Omega$

or

$\nabla^2u=f$ in $\Omega$
$u=0$ on $\partial \Omega$ ?

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• This is a very broad question. Are you asking about the study of eigenvalues of discretizations for purposes of numerical stability or for purposes of understanding "physics"? – David Ketcheson Jan 23 '12 at 20:52
• @DavidKetcheson: I edited the question and provide an example for context. Does this make it any more clear? – Paul Jan 24 '12 at 15:50
• I think this is purely a math question and should be migrated to math.SE. I don't see how the word "numerically" enters into the question at all. Would you agree? – David Ketcheson Jan 24 '12 at 16:34
• @DavidKetcheson: yeah... I agree with you. Perhaps it would be best submitted to the math.se. Is there a way I can migrate it myself? – Paul Jan 24 '12 at 17:46
• Eigenvalue problems usually come from separation of variables. If you try solutions of the form f(x)g(t) of a PDE that depends on x and t you get an eigenvalue equation for f and another eigenvalue equation for g. The general solution of your original PDE is then a linear combination of those products, summed over all possible values for the eigenvalue. – Jules Apr 12 '18 at 11:22

For your specific question, observe that $$\nabla^2 u = 0$$ is a special case of the eigenvalue equation $$\nabla^2 u = \lambda u$$ with $\lambda = 0$. So understanding the eigenvalue problem certain helps understanding the homogeneous problem.

For the inhomogeneous problem, formally if one knows all the possible eigenvalues $\lambda_k$ and corresponding eigenfunctions $u_k$, then one can try to write $$f = \sum_{k} c_k u_k$$ as a combination of the eigenfunctions with constant coefficients $c_k$. If this is possible, then we can solve $$\nabla^2 u = f$$ by formally inverting the operator to get $$u = \sum_{k} \frac{c_k}{\lambda_k} u_k$$ And so yes, understanding the eigenvalues and eigenfunction can also help understand the inhomogeneous linear problem.

In quantum mechanics, eigenvalues are of physical interest. The eigenvalues of the Hamiltonian operator (which determines how the wavefunction propagates in time) are also the only allowed energies of the system.

When propagating in time, certain physical effects can be simulated by using an incomplete eigenvalue basis. To simulate a "core" electron of a big atom like Xenon, one can make the assumption that the outer electrons remain where they are and leave their energy states out of the core state's basis set to simulate the effect of the exchange interaction on the core. One can do the same with the outer and core electrons switched.

Selection rules allow the bandwidth of eigenvalue bases to be lower than that of finite difference or finite element matrices.

For a classical elastic system, the eigenvalues and eigenstates correspond to the natural frequencies and normal modes of the system. The system will tend to oscillate at these frequencies, so one must take care not to couple the system to vibrations at these frequencies.

• I like this answer because you show that eigenvalues and eigenvectors are of interest in their own in physics. These applications are fascinating. – shuhalo Aug 9 '17 at 0:49

Solving a time dependent equation like the heat equation $$u_t-\nabla^2u=0,\quad x\in\Omega,\quad t>0$$ with boundary condition $u(x,t)=0$ if $x\in\partial\Omega$, $t>0$, and initial condition $u(x,0)=u_0(x)$, $x\in\Omega$, by separation of variables (that is, looking for solutions of the form $u(x,t)=T(t)X(x)$) leads in a natural way to the eigenvalue problem. The solution can be expanded (under some conditions on $\Omega$ and $U_0$) as $$u(x,t)=\sum_{n}e^{-\lambda_nt}X_n(t),$$ where $\lambda_n$ are the eigenvalues and $X_n$ the associated eigenfunctions.