The particularity of $k$ being an integer in the solution of a DE

Question

Related to the Thomas's comment in the question : Eigenvalues of the circle over the Laplacian operator, is there anyone could tell me why the periodic function $g$ has a fundamental set of solutions of the form $c_1 \cos (\mu x) + c_2 sin (\mu x)$, where $\mu \in \mathbb{Z}$? Someone told me that is related to the Laplace equation and the boundary condition $u(0)=u(2 \pi)$ $u'(0)=u'(2 \pi)$ but it is very unclear for me.

Answer 1

Essentially, the problem is $u''+\lambda u=0$ on $[-\pi,\pi]$ with periodic boundary conditions. (Note that this interval has the same length as the circumference of the circle; this choice of parametrization ensures that the Laplace-Beltrami operator on the circle directly corresponds to the Laplacian on this interval.) The solutions to the DE itself are sinusoids with angular frequency $\lambda$. The boundary conditions can be satisfied if and only if $\lambda$ is a nonnegative integer (so that $2 \pi$ is a period).

Edit: you need $\sin(\lambda(-\pi))=\sin(\lambda(\pi))$ and $\cos(\lambda(-\pi))=\cos(\lambda(\pi))$. If you let $-\lambda \pi=x$ then you need $\sin(x)=\sin(x+2\pi\lambda)$ and $\cos(x)=\cos(x+2\lambda \pi)$, which is true if and only if $\lambda$ is an integer. Now $\lambda$ also had to be nonnegative in order to get the sinusoid solutions in the first place, so the admissible $\lambda$ are the nonnegative integers. (This means the eigenvalues in the usual sense are the nonpositive integers.)