This paper concerns the vertex reinforced jump process (VRJP), the edge reinforced random walk (ERRW), and their relation to a random Schrödinger operator. On infinite graphs, we define a 1-dependent random potential β \beta extending that defined by Sabot, Tarrès, and Zeng on finite graphs, and consider its associated random Schrödinger operator H β H_\beta . We construct a random function ψ \psi as a limit of martingales, such that ψ = 0 \psi =0 when the VRJP is recurrent, and ψ \psi is a positive generalized eigenfunction of the random Schrödinger operator with eigenvalue 0 0 , when the VRJP is transient. Then we prove a representation of the VRJP on infinite graphs as a mixture of Markov jump processes involving the function ψ \psi , the Green function of the random Schrödinger operator, and an independent Gamma random variable. On Z d {\Bbb Z}^d , we deduce from this representation a zero-one law for recurrence or transience of the VRJP and the ERRW, and a functional central limit theorem for the VRJP and the ERRW at weak reinforcement in dimension d ≥ 3 d\ge 3 , using estimates of Disertori, Sabot, and Tarrès and of Disertori, Spencer, and Zimbauer. Finally, we deduce recurrence of the ERRW in dimension d = 2 d=2 for any initial constant weights (using the estimates of Merkl and Rolles), thus giving a full answer to the question raised by Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schrödinger operator H β H_\beta .