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 〈1, V1〉 ≠ 0. Spectral properties of H in the low-energy limit are analyzed. Asymptotic expansions for R(ζ) = (H – ζ)–1 and the S-matrix S(λ) are deduced for ζ → 0 and λ ↓ 0, respectively. Depending on the zero-energy properties of H, the expansions of R(ζ) take different forms. Generically, the expansions of R(ζ) do not contain negative powers; the appearance of negative powers in ζ1/2 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