  We first show that infinite satisfiability can be reduced to finite
satisfiability for all prenex formulas of Separation Logic with $k\geq1$
selector fields ($\seplogk{k}$). Second, we show that this entails the
decidability of the finite and infinite satisfiability problem for the class of
prenex formulas of $\seplogk{1}$, by reduction to the first-order theory of one
unary function symbol and unary predicate symbols. We also prove that the
complexity is not elementary, by reduction from the first-order theory of one
unary function symbol. Finally, we prove that the Bernays-Sch\"onfinkel-Ramsey
fragment of prenex $\seplogk{1}$ formulae with quantifier prefix in the
language $\exists^*\forall^*$ is \pspace-complete. The definition of a complete
(hierarchical) classification of the complexity of prenex $\seplogk{1}$,
according to the quantifier alternation depth is left as an open problem.
