The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-(2\pi/L)^2 n^2$.
On the other hand the Laplacian of a non-compact manifold may be continuous. For example on $\mathbb{R}$ the spectrum of the second derivative operator is $(-\infty,0]$.
I was wondering if it is always the case that when a sequence of pointed Riemannian manifold $(M_n,o_n,g_n)$ converges smoothly to a limit Riemannian manifold $(M,o,g)$ then the spectrum of the corresponding Laplace operators also converges. I don't think this is true but a semi-continuity property should hold (and I'm guessing is well known) but I'm having trouble finding a reference.
As far as I understand the eigenvalues satisfy domain monotonicity, which means that a open submanifold with boundary has eigenvalues which are larger in absolute value then the whole manifold (though I don't really understand what boundary conditions should be imposed). If, in fact, the first eigenvalue on the whole manifold is the limit of first eigenvalues on an increasing sequence of subdomains then this is enough to prove upper semi-continuity of the absolute value of the first eigenvalue of the Laplacian under smooth convergence.
Assuming that putting a metric on $\mathbb{R}^2$ which is flat on a half-plane and hyperbolic on the other half gives a non-zero first eigenvalue to the Laplacian one would obtain a counterexample to continuity. This is because moving the basepoint to infinity in the flat half-plane the sequence converges smoothly to flat $\mathbb{R}^2$ (for which the first eigenvalue is $0$).
To start I'd like to know if the above reasoning is correct and if there is a good reference. Concretely I'd love to have a good reference for the answers to the following questions:
- Is the absolute value of the first eigenvalue of the Laplacian upper semi-continuous with respect to smooth convergence?
- Is there a simple counterexample for continuity?
The question: What about the other eigenvalues? isn't definite enough for this site so I'll try to make it more concrete.
By the spectral theorem the (unique self-adjoint extension of the) Laplacian on the limit manifold is unitarily conjugate to multiplication by some (non-positive) function $\varphi$ on $L^2(\mu)$ for some $\sigma$-finite measure $\mu$ on a Polish space $X$. Let $\varphi_n,\mu_n$ and $X_n$ be defined similary for the sequence.
Is it true that $\mu(\varphi^{-1}((-a,0])) \ge \limsup \mu_n(\varphi_n^{-1}((-a,0]))$ for all $a > 0$?