  In this paper we study Lagrangian tori in ${\mathbb C}P^2$. A two-dimensional
periodic Schr\"odinger operator is associated with every Lagrangian torus in
${\mathbb C}P^2$. We introduce an energy functional for tori as an integral of
the potential of the Schr\"odinger operators, which has a natural geometrical
meaning. We study the energy functional on two families of Lagrangian tori and
propose a conjecture that the minimum of the functional is achieved by the
Clifford torus. We also study deformations of minimal Lagrangian tori. In
particular we show that if the deformation preserves a conformal type of the
torus, then it also preserves the area of the torus. Thus it follows that
deformations generated by Novikov-Veselov equations preserve the area of
minimal Lagrangian tori.
