I'm reading a book about solving pde's and the following text has come up:
In analysing the convergence of iterative solution methods in subsequent chapters, we will make extensive use of spectral properties of the discrete Laplace matrices. To derive these properties we resort to the continuous counterpart and consider the related Sturm-Liouville problem. This means that in case of the one-dimensional problem we look for the eigenvalues and eigenfunctions $u\neq 0$, such that $$-\dfrac{d^2u(x)}{dx^2} = \lambda u(x)$$ supplied with homogeneous Dirichlet or Neumann boundary conditions.
In case of Dirichlet boundary conditions $$u^{[k]}(x) = \sin(k\pi x)\text{ corresponding to }\lambda_k = k^2\pi^2\text{ for } k\in\mathbb{N},k\neq 0$$ (imposing $k\neq 0$ is required to assure a non-trivial eigenfunction)
In case of Neumann boundary conditions$$u^{[k]}(x) =\cos(k\pi x)\text{ corresponding to }\lambda_k = k^2\pi^2 \text{ for }k\in\mathbb{N} $$ ($k = 0$ gives the constant eigenvector)
I am trying to understand what this (globally) means, and I'm a bit uncomfortable with the notation that is used.
Question: What does the superscript $[k]$ mean in $u^{[k]}$? What is $u^{[k]}$ here?
The $$ -\frac{d^{2}u}{dx^{2}} = \lambda u(x)$$ has non-trivial solutions. For example, $ u(x) = \sin(x) $ is a solution, with $\lambda = 1$. Another is $u(x)= \cos(2x)$, with $\lambda = 4$. $$ u^{[k]}(x) = \sin(k \pi x)$$ is the solution with $\lambda= \lambda_{k} =( k \pi)^{2}$. The author uses $u^{[k]}(x)$, i can also write it as $u_{k}(x)$. But using notation $u^{k}(x) $ could create ambiguity with $u(x)^{k}$.