How guarantee that the Sturm-Liouville spectrum is discrete? Suppose that we want to series expand a function $f \in L^2_w(0,\infty)$ in terms of eigenfunctions of a given Sturm-Liouville system having weight function $w$ for the half line, and that we have already found that the left end ($0$) is regular, while the right end ($+\infty$) is limit point.
What would be a tick list of methods/theorems we can use for checking that the essential spectrum (i.e., the continuous spectrum that is not also part of the point spectrum) is empty, so that we can avoid an integral corresponding to the essential spectrum?  If nonempty, could we impose additional "reasonably practical" conditions at $+\infty$ which would ensure that the essential spectrum is empty?
 A: Suppose that the equation is
$$ - (p(x) y'(x))' + q(x) y(x) = \lambda w(x) y(x)$$
defined on $[0,\infty)$.
First of all, at an LP endpoint, a boundary condition is neither
required nor allowed (Zettl [1, Sect. 7.1]).  Second, the classification
of an LP endpoint at $\infty$ depends only on the asymptotic behavior
of $1/p, q, w$ (Zettl [1, Sect. 7.2]), so the LP classification itself
is difficult to escape.
However, a viable approach is to try to prove that the spectrum is
discrete as is. A simple method is presented by Titchmarsh [2, Ch. V]:
First, apply the Liouville transformation (if possible; see, e.g.,
Everitt [3, 4]) to transform the equation to the form $$ - y'' + Q y =
\lambda y .$$ If $Q(x) \to \infty$ when $x \to \infty$, then
the $\infty$ endpoint is LP and the spectrum is discrete.
When the ultimate purpose is to find a series expansion, it may be
desirable not only to prove that the spectrum is
discrete, but also bounded from below. This has been called the BD
property, and is discussed briefly in Zettl [1, Sect. 10.13].
Somewhat surprisingly, there is a simple sharp condition for the BD
property discovered by Molchanov. The Liouville-transformed system 
(i.e., $p=w=1$) has
the BD property if and only if, for some $h, 0 < h < \infty$,
$$\int_t^{t+h}Q(x)\,dx \to \infty$$
when $t \to \infty$.
[1] Zettl, A.: Sturm-Liouville Theory. AMS. 2005.
[2] Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-order
Differential Equations. Part I. 2nd ed. Oxford at the Clarendon press. 1962.
[3] Everitt, W.N.: Charles Sturm and the Development of Sturm-Liouville
Theory in the Years 1900-1950.
In Amrein et al. (eds.): Sturm-Liouville Theory: Past and Present.
Birkhäuser Verlag. 2005.
[4] Everitt, W.N.: A Catalogue of Sturm-Liouville Differential Equations. In
Amrein et al. (eds.): Sturm-Liouville Theory: Past and Present.
Birkhäuser Verlag. 2005.
