I’m interested in using laplacian $(-∆)$ eigenfunction as a basis for $H^1 (R^n )$. I know that in $H^1 (Ω)$, $Ω$ bounded this can be done so I was wandering about $H^1 (R^n )$.
Now let $e_λ$ be an eigenfunction corresponding to $λ$ so that: $-∆e_λ=λe_λ$
In addition, consider that $⟨*,*⟩$ is the scalar product on $H^1 (R^n)$
Here is what I know:
1.The spectrum of $-∆$ is $(0,∞)$
2.We have the representation: $∀u∈H^1 (R^n )$ $u(x)=∫_0^∞⟨u,e_λ ⟩e_λdλ ∀u∈H^1 (R^n )$
3.In general, if $H$ is a hilbert space ( with a countable or uncountable basis) any element of $H$ can be represented as an infinite series. Namely, if $M$ si a countable or uncountable set and ${T_μ |μ∈M}$ is a base for $H$ then: $∀u∈H$ $∃N⊂M,N$ countable such that: $u=∑_{n∈N}c_n T_n$
First thing first. Do we actually have eigenfunctions of Laplacian in $H^1 (R^n )$ ?. It is clear that $sin(x_1)$ satisfies the PDE but is not square integrable. And if we don’t have then I guess that spectral theory can’t be of use with $∆$ and $H^1 (R^n)$ right ?
II Assuming that we actualiy have $e_λ∈H^1 (R^n )$ Are the statement 1, 2, 3 correct ? I’m confident in 1 and 3 but I want to check.
III Does 2 imply some type of density of $span{e_λ |λ∈(0,∞)}$ in $H^1 (R^n )$ ?
IV. can we combine 2 and 3 to get $∀u∈H$ $∃N⊂(0,∞),N$ countable such that: $u=∑_{n∈N}c_n e_n$ ? Basically, for a specific element of $H$ can we chose to work with the infinite sum and not the integral ?
