  We introduce and analyze the new range-separated (RS) canonical/Tucker tensor
format which aims for numerical modeling of the 3D long-range interaction
potentials in multi-particle systems. The main idea of the RS tensor format is
the independent grid-based low-rank representation of the localized and global
parts in the target tensor which allows the efficient numerical approximation
of $N$-particle interaction potentials. The single-particle reference potential
like $1/\|x\|$ is split into a sum of localized and long-range low-rank
canonical tensors represented on a fine 3D $n\times n\times n$ Cartesian grid.
The smoothed long-range contribution to the total potential sum is represented
on the 3D grid in $O(n)$ storage via the low-rank canonical/Tucker tensor. We
prove that the Tucker rank parameters depend only logarithmically on the number
of particles $N$ and the grid-size $n$. Agglomeration of the short range part
in the sum is reduced to an independent treatment of $N$ localized terms with
almost disjoint effective supports, calculated in $O(N)$ operations. Thus, the
cumulated sum of short range clusters is parametrized by a single low-rank
canonical reference tensor with a local support, accomplished by a list of
particle coordinates and their charges. The RS canonical/Tucker tensor
representations reduce the cost of multi-linear algebraic operations on the 3D
potential sums arising in modeling of multi-dimensional data by radial basis
functions, say, in computation of the electrostatic potential of a protein, in
3D integration and convolution transforms, computation of gradients, forces and
the interaction energy of a many-particle systems, and in low parametric
fitting of multi-dimensional scattered data by reducing all of them to 1D
calculations.
