I have a question about a statement I came across which I'd be happy to understand more.
On $L^2(0,1)$, we can consider two self-adjoint operators. The first operator $H_0$ acts as $H_0f=-f''$, with domain consisting of all $H^1$ functions satisfying $f(0)=0$ and $f'(1)=f(1)$. The second operator $H_1$ acts as $H_1f=-f''$, with domain consisting of all $H^1$ functions satisfying $f(0)=0$ and $f'(1)=-f(1)$.
On numerous occasions I have encountered statements similar to - "$H_1$ is a finite rank perturbation of $H_0$" (sadly I can't provide sources...). I've seen this statement on all different kinds of changes in boundary conditions (when the boundary is finite).
My question is - in what sense is this a finite rank perturbation? Usually when one says this, they mean that the difference is of finite rank. But here, the domains of the operators are different, and so the difference naively doesn't make sense. So in what sense is this true? I have a feeling it is related to the associated quadratic form, but I'm not sure.
Thanks a lot in advance.
