Consider the eigenvalue problem Δφ=λφ with the Dirichlet boundary condition on the rectangle Ω=[0,1]×[0,1] .
By using separation of variable method { φ(x,y)=f(x)g(y) }, I found that φ(x,y)=sin(mπx)sin(nπy) for m,n≥1 with λ=π^2(m^2+n^2) satisfies the equation so that they are one of the eigenvalues and eigenfunctions.
But how do we know that they are in fact all the eigenvalues and eigenfunctions? Is there any possibility that there exists some eigenfunction that is not separable (so that it cannot be found by separation of variable method) but satisfies the equation?
Another eigenfunction has either a different eigenvalue, not in the spectrum, then its orthogonal to this complete basis.
Or it has one of eigenvalues in the spectrum, than it has to be orthogonal to the 2d space of solutions of ordinary eigenvalue differential equations with the given boundary condtions.
Since spectrum and basis are a four parameter space over the boundary conditions $a f(0)+b f'(0)=0$b , its essential to fix them.
Eg $f(0))=0$ and $f'(1)=0$ defines a competely different operator $\Delta$ on $(0,1)$.