exact behavior of Jost functions at low-energy
For Schrödinger operators with central potential q(r) and angular momentuml, the behavior of the Jost function F l (k) as k→0 is investigated. It is assumed that ∫∞ 0 d r (1+r)σ‖q(r)‖<∞, where σ≥1. Situations where q is integrable with 1≤σ<2, but not with σ≥2 are of particular interest. For potentials satisfying q(r)∼q 0 r −2−ε (0<ε≤1) and l=0, the leading behavior of F 0(k) and the phase shift δ0(k) as k→0 is derived. Also comments are made on the differentiability properties of the Jost solutions with respect to the variable k at k=0. For σ=1 Levinson’s theorem is proved, thereby clarifying some questions raised recently by Newton [J. Math. Phys. 2 7, 2720 (1986)].