Variational formulas and approximation theorems for the first eigenvalue in dimension one (Q1609653)

On a finite or semi-infinite interval the author considers a general superposition \(L\) of the first and second derivative with \(x\)-dependent coefficients and with Dirichlet or Neumann boundary condition at the origin. The work is a continuation of its seven (all self-) references [this makes it less easy to put this paper in broader context since just one of them (but Trans. Am. Math Soc.!) is extra-territorial] but offers very nice results (their essence being well characterized by the title, and they look ``final''). Amazingly enough, these explicit bounds are complete (i.e., both-sided)! Three illustrative examples demonstrate their power in applications. The study is complemented by its discrete, Markov-chain parallel considering the birth-death process mediated by the purely second-order difference operator \(D\) (personally, I would recommend to read this supplement first).