While looking at the Hilbert–Pólya conjecture , I thought of the following question which I couldn't answer:
Given a real, monotonically increasing (and divergent) sequence $\left\{\lambda_n\right\}_{n=1}^\infty$, under what conditions can we construct some domain and boundary conditions such that the eigenvalues of the Laplacian on this domain are exactly the series' elements (ideally with multiplicities)?
It seems like to define this question well the domain would need to be some Riemannian manifold, and I've only seen sources dealing with strictly Dirichlet or strictly Neumann boundary condition.
Specifically since this deals with the Hilbert–Pólya conjecture, the series in question is the imaginary parts of the Zeta function zeros for which the number of elements less than $T$ is $\frac{T}{2\pi}\log(\frac{T}{2\pi e})+O\left(\log(T)\right)$. So if there is anything that can be said for that particular case and not generally it would be great to know about that.
A few things I do know:
- Due to Weyl's law, it seems unless the series diverges as some power law asymptotically, the domain cannot have finite nonzero volume, at least if the domain is well behaved and uses Dirichlet boundary conditions
- This question on mathoverflow seems to state that for the Schrödinger operator there are situations in which the number of elements less than $T$ diverges like $T^\frac{d-1}{2}\ln(T)$, so with $d=3$ this is exactly the Hilbert-Pólya case
- There seems to be many more things that can be said about a manifold from its eigenvalues, which would seem to add more limitations on $\lambda$.