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Let N(T ; V) denote the number of eigenvalues of the Schrödinger operator −y ′′ + Vy with absolute value less than T . This paper studies the Weyl asymptotics of perturbations of the Schrödinger operator −y ′′ + 1 4 e 2t y on [x 0 , ∞). In particular, we show that perturbations by functions ε(t) that satisfy |ε(t)| e t do not change the Weyl asymptotics very much. Special emphasis is placed on connections to the asymptotics of the zeros of the Riemann zeta function.

In this paper, we continue some recent work on two weight boundedness of sparse operators to the "off-diagonal" setting. We use the new "entropy bumps" introduced in by Treil-Volberg ([21]) and improved by Lacey-Spencer ([9]) and the "direct comparison bumps" introduced by Rahm-Spencer ([19]) and improved by Lerner ([10]). Our results are "sharp" in the sense that they are sharp in various particular cases. A feature is that given the current machinery, the proofs are almost trivial.

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