The authors consider the linear differential equation \(\mathcal{L}x\equiv -x''+a(t)x=0\), \(t\geq T\geq 0\), where \(a(t)\) is a continuous, complex-valued function. Using the Cauchy-Schwarz inequality, they prove the nonexistence of nontrivial solutions square integrable on an infinite interval under milder assumptions on the coefficient \(a(t)\). This, in particular, means that the operator \(\mathcal{L}\) is of the limit-point type at infinity. The authors also discuss extensions of the result to third-order equations.