A Comparison Theorem for Hamiltonian Vector Fields

The question of completeness of Hamiltonian systems is investigated for a class of potentials not necessarily bounded below. The result generalizes previous work of W. Gordon and D. Ebin. This paper extends the completeness theorem of Ebin [1] to include certain potential functions V not necessarily bounded below. The condition on V is essentially the same as a condition for a corresponding quantum mechanical theorem. See [3] and Remark 3 below. We shall prove a comparison theorem which reduces the general case to the one-dimensional case, so we begin with the latter. 1. The one-dimensional case. Let R+ be the nonnegative reals and Vo:R+~R a nonincreasing Cl function. Consider the Hamiltonian system with the usual kinetic energy and potential Vo; i.e. if c(t) is a solution curve we have dVo c"(t) = (c(t)). dx By monotonicity of Vo, if c'(O) i?;0 then c'(t) i?;0 for all ti?;O. Thus if H = [c'(t)/2 ]2+ Vo(c(t)) is the constant total energy, c'(t) = 2(H Vo(c(t)))l/2. DEFINITION. The potential Vo is positively complete iff f~ (2(H _ d;O(X)))l/2 ~ 00 as x ~ 00 for all Xli?;O and H such that VO(Xl) <H. I t is easy to see that if this holds for some Xl, H, such that VO(Xl) <H then it holds for all such Xl, H (use the fact that improper integrals with asymptotic integrands are simultaneously convergent or divergent). Received by the editors March 15, 1970. AMS 1969 subject classifications. Primary 3465; Secondary 3442.