On Littlewood’s Counterexample of Unbounded Motions in Superquadratic Potentials

Littlewood [11] constructed an example of an equation of the form ẍ+ V’(t) with $$ \frac{{V'(x)}}{x} \to \infty $$ and yet possessing unbounded solutions. This note contains a considerable simplification of Littlewood's construction together with rather precise information on the behavior of $$ \frac{{V'(x)}}{x} \to \infty $$ under which the resonances of the type constructed by Littlewood are possible. The original construction [11] contained an error, as was pointed out by Yiming Long, who also corrected the original proof [12].