# Simple eigenvalue for eigenfunctions

I know what it means that an eigenvalue of a matrix is simple. My question is about the eigenvalues of operators as the Schröndiger one. Here we work with eigenfunctions rather than with eigenvectors so I can not see what the algebraic multiplicity is.

Take, for example, the one dimensional harmonic oscillator. The Schrodinger equation is - $$\left (-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac{1}{2}m\omega x^2 \right )\psi=E\psi$$
And the corresponding energies are - $$E_n=\hbar \omega\left (n+\frac{1}{2}\right ), n\in\mathbb N$$
In this case, since each value of $$n$$ corresponds to a distinct eigenfunction, we know that each energy (eigenvalue) has only one corresponding eigenfunction, and the eigenvalues are thus all simple. This is unlike the case of the $$2$$ or $$3$$ dimensional harmonic oscillator, which can have degenerate states - energies with certain multiplicity.