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
.