Has uniform ellipticity implications on the spectrum? Let $X$ be a complete Riemann surface with a smooth metric, and $L$ a line bundle on it also equipped with a smooth metric; associated to this data there is a Laplace-Beltrami operator $D_L$ acting on sections of $L$.
We know that if $X$ is compact the operator $D_L$ has only discrete spectrum.
In the specific situation I'm considering $X$ is not compact but the metrics are such that $D_L$ is uniformly elliptic, and not just elliptic as in the general non-compact case. I would like to conclude from this that $D_L$ is discrete, viewing the uniform ellipticity as a "replacement" for the compactness of the surface.
Does it work? If it does not, is it possible to add some more hypothesis to deduce such a result or the facts


*

*$D_L$ is uniformly elliptic on a complete surface

*$D_L$ is discrete


are completely unrelated? 
 A: I tend to consider such things from the perspective of Dirac operators. 
Let $D$ be a Dirac operator on a 
complete Riemann manifold, acting on a vector bundle $E$. 
There is a Weitzenboeck type formula 
$D^2 = \nabla^{\ast} \nabla + R$. The Weitzenboeck remainder $R$ is of order zero and symmetric. We write $\nabla^* \nabla =: \Delta$, the Laplace
operator. The Laplace you are interested in is $D^2$, not $\Delta$. 
It is a standard result that $D^2$, $D$ and $\Delta$ are essentially selfadjoint.
If you take the operator $D=d+d^{*}$ on exterior 
forms, suitably twisted with a line bundle, you get the situation you were takling about in
the question.
You want to know a criterion for $D$ or equivalently $D^2$ to have a discrete spectrum. 
I believe that the following results are true, and indicate a proof (potentially flawed). The basic result is the following extension of the Fredholm property on noncompact manifolds.
THEOREM 1: ''Let $\Delta =\nabla^{\ast} \nabla$. Let $P$ be an essentially selfadjoint elliptic operator
of order $2$.
Assume that there exists $c>0$ and a compact set $K \subset M$ such that for all compactly supported
sections $u$ with support disjoint from $K$, one has 
$$\langle Pu,u \rangle \geq \langle \Delta u , u \rangle + c^2 \langle u, u\rangle. $$
Then $P$ is Fredholm.''
I defer the proof to the end of this post, as it is technical, and discuss some applications. 
COROLLARY 1: ''Let $D$ be a Dirac operator on a complete manifold $M$. Assume that there exists a compact 
$K$ and a $c>0$ such that the Weitzenboeck remainder $R$ is estimated from below by $c^2$ over $M-K$. Then 
$spec (D) \cap (-c,c)$ is discrete.''
One could phrase this by saying that an estimate of the Weitzenboeck remainder from below enforces an essential
spectral gap, at least as large as $c$.
PROOF:
Let $\lambda^2 < c^2$. Let $Q := D^2-\lambda^2$. Over $M-K$, we find that $Q = \Delta + R-\lambda^2 \geq \Delta + (c^2-\lambda^2)$.
By Theorem 1, $Q$ is Fredholm. Thus $\lambda^2$ is an isolated point of the spectrum of $D^2$, and hence $\pm \lambda$
are isolated points of $spec (D)$. QED
Note that in Corollary 1, the size of $K$ was inessential and could be increased, which allows the possibility for a stronger estimate on the complement of $K$ and hence
to enlarge the essential spectral gap. This is the content of the next result.
THEOREM 2: ''Let $D$ be a Dirac operator on $M$ and $f: M \to \mathbb{R}$ be a proper function 
that is bounded from below. If the remainder term $R$ in the Weitzenboeck formula $D^2 = \Delta + R$ is bounded
below by $f$, then the spectrum of $D$ (and hence that of $D^2$) is discrete.''
PROOF: For each $C>0$, we find a compact set $K \subset M$ and an estimate $R \geq C^2$ on $M-K$. By Corollary 1, 
$spec (D) \cap (-C,C)$ is discrete. As $C$ was arbitrary, $spec (D) \subset \mathbb{R}$ is discrete. QED.
PROOF OF THEOREM 1:
The first step is a strengthening of Gardings inequality. Namely, we prove that there is a constant $C$ with 
$$
(!!!) \; \|u \|_2 \leq C (\| u|_K \|_0 + \| Pu \|_0).
$$
Without loss of generality $c \leq 1$. 
Let $u$ be a section with compact support 
$supp (u)\subset M-K$. I write $\| u \|_r$, $r=0,1,2$, for the Sobolev norms.
Estimate using the condition on $P$:
$$
\| u \|_2^2 = \|u \|_0^2 + \|\Delta u \|_0^2 \leq \frac{1}{c^2} \|P u\|_0^2 
$$
If $K \subset U$ is a relatively compact neighborhood, and $u$ supported in $U$, the usual Garding inequality
shows 
$$
\| u \|_2^2 \leq C(\| u\|_0^2 + \| Pu\|_0^2)
$$
To patch both estimates together, choose $a$ supported in $U$, $b$ supported in $M-K$ such that $a + b =1$.
For an arbitrary compactly supported section $u$, we find that
$$
\| u \|_2 \leq \| au \|_2+\| bu \|_2 \leq C' (\| au\|_0^2 + \| P a u\|_0^2 + \| Pbu\|_0^2) $$
The first term is bounded by $\| u|_K \|_0 $. 
Furthermore $\| P a u\|_0 \leq \| [P,a] u \|_0 + \| a P u \|_0 \leq  \| [P,a] u \|_0 + C\|  P u \|_0$. But $[P,a]$ has order $1$ and is supported in $U$. Therefore, there is a function $h$ with $[P,a]h=[P,a]$ and hence $\| [P,a] u \|_0 \leq C \| hu \|_1 $. By the Peter-Paul inequality, there is for $\epsilon>0$ a constand $C(\epsilon)$ such that $\| hu \|_1  \leq \epsilon \| hu \|_2 + C(\epsilon) \| hu \|_0\leq C\epsilon \| u \|_2 + \| u|_K\|_0$. Picking $\epsilon$ small enough, we can swap the first summand to the left hand side of the estimate. Similarly, the term $\|Pbu\|$ is estimated, and we have proved the strengthened inequality (!!!).
Why did I do all this stuff? Well, the operator $W^2 \to L^2 \stackrel{|_K}{\to} L^2$ is compact by Rellichs theorem. It is a general fact (e.g. Proposition 3.7.2 in my lecture notes http://wwwmath.uni-muenster.de/u/jeber_02/skripten/mainfile.pdf) that this implies that $P$ has closed range and finite-dimensional kernel. Since $P$ is essentially 
selfadjoint, the cokernel is the kernel of $P$ and so finite-dimensional as well. QED (Theorem 1).
