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 $-\lambda\_n = -(2\pi/L)^2 n^2$.


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 use the convention that the Laplacian is negative semi-definite).


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, hence the reference-request tag). 


Given a Riemannian manifold $M$ let $-\lambda\_1(M)$ be the maximum of the spectrum (i.e. the closest point to $0$). Then $\lambda\_1$ satisfies the following "domain monotonicity": If $\Omega$ is an open submanifold of $M$ with boundary then $\lambda\_1(\Omega) \ge \lambda\_1(M)$ (e.g. $\lambda\_1(\mathbb{R})=0$ but $\lambda\_1([0,2\pi]) = 1$). If one knows that $\lambda\_1(M)$ is the limit of $\lambda\_1(\Omega\_n)$ over an increasing sequence of bounded submanifolds then this is enough to prove upper semi-continuity of the absolute value of $\lambda\_1$ under smooth convergence.


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:


1. **Is $\lambda\_1$ upper semi-continuous with respect to smooth convergence?**
2. **Is there a simple counterexample for continuity?**


The question: **What about the rest of the spectrum?** 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 $a > 0$?**


**Edit: (Hopefully) Fixed some issues (positive vs negative definite Laplacian, first eigenvalue vs maximum of the spectrum) in response to comments.**



Behavior of the spectrum of the Laplacian under pointed smooth convergence
This is only a partial answer since it concerns only compact manifolds. There is a result due to T. Shioya <http://projecteuclid.org/download/pdf_1/euclid.jmsj/1213023969> a special case of which says that if a sequence of compact $n$-dimensional Riemannian manifolds $M\_i$ converges to another compact n-dimensional Riemannian manifold M in the Gromov-Hausdorff sense and such that the sectional curvature is uniformly bounded from below, then for any $k$ the $k$-th eigenvalue of the Laplacian of $M\_i$ converges to the $k$-th eigenvalue of the Laplacian of $M$. Notice that smooth convergence implies the above type of convergence.


