What are the solutions to the equation
uxx + uyy = λu
on the unit square with the requirement that u(x, y) = 0 on the boundary?
It’s easy to see that the functions
u(x, y) = sin(mπx) sin(nπy)
are solutions with
λ = (m² + n²)π²
for non-negative integers m and n. It’s not so obvious that these are the only solutions, but we’ll take that on faith.
The previous post looked at Pólya’s bounds on the eigenvalues of the Laplace operator Δ for a general region D, with only the requirement that copies of D can tile the plane without overlapping. Surely squares satisfy this requirement, so the problem in this post is a special case of the problem in the previous post. So how do they compare?
Pólya gives lower bounds on the kth eigenvalue, so how do we order the numbers of the form (m² + n²)π?
There’s a theorem for counting the number of numbers n up to x where n is the sum of two squares, the Landau-Ramanujan theorem. But it seems to contradict Pólya’s bounds. That’s because Landau-Ramanujan only counts each n once, but eigenvalues need to be counted multiple times if a number can be written as a sum of squares multiple ways.
For example, 25π² should count as an eigenvalue four times, corresponding to (m, n) = (5, 0), (0, 5), (3, 4), and (4, 3).
About how large is the kth eigenvalue? If non-negative integers m and n satisfy
(m² + n²)π² ≤ x
then (m, n) is inside a circle of radius √x/π. For large x, the number of such pairs is approximately the area of a circle of radius √x/π in the first quadrant, which is x / 4π. So the kth eigenvalue is approximately 4πk, which matches Pólya’s lower bound of 4πk for a region of area 1.
The post Eigenvalues of the Laplacian on a square first appeared on John D. Cook.