Spectral properties in the low-energy limit of one-dimensional Schrödinger operators H = –d2/dx2 + V. The Case <1, V1> =? 0

In this paper we consider the Schrödinger operator H = –d2/dx2+ V in L2(R), where V satisfies an abstract short-range condition and the (solvability) condition =? 0. Spectral properties of H in the low-energy limit are analyzed. Asymptotic expansions for R(?) = (H – ?)–1 and the S-matrix S(?) are deduced for ? ? 0 and ? ? 0, respectively. Depending on the zero-energy properties of H, the expansions of R(?) take different forms. Generically, the expansions of R(?) do not contain negative powers; the appearance of negative powers in ?1/2 is due to the possible presence of zero-energy resonances (half-bound states) or the eigenvalue zero of H (bound state), or both. It is found that there are at most two zero resonances modulo L2-functions