  In this paper we derive and analyse new exponential collocation methods to
efficiently solve the cubic Schr\"{o}dinger Cauchy problem on a $d$-dimensional
torus. Energy preservation is a key feature of the cubic Schr\"{o}dinger
equation. It is proved that the novel methods can be of arbitrarily high order
which exactly or nearly preserve the continuous energy of the original
continuous system. The existence and uniqueness, regularity, global
convergence, nonlinear stability of the new methods are studied in detail. Two
practical exponential collocation methods are constructed and two numerical
experiments are included. The numerical results illustrate the efficiency of
the new methods in comparison with existing numerical methods in the
literature.
