Integrable Potentials and Half-line Spectra1

is not assumed. Actually, if (3') holds, much more is known. In fact, in this case, there exist asymptotic formulas for the solutions of (1) whenX>0 ([8, p. 421]; cf. also [7, p. 258], in case/(<)—>0 as 2->oo) which guarantee, in particular, that 0^Xoo, as T—kx>, and, in fact, even with the requirement that foT\f(t) I dt—* oo arbitrarily fast. Thus, if (t) denotes any positive function satisfying #(/)—>c0 as /—>oo, there exists a continuous function f(t) on 0g*(T) =o(f\f0T(t)\dt), as r->oo.