We investigate approximations by finite sums of products of functions with separated
variables to eigenfunctions of certain class of elliptic operators in higher dimensions,
and especially conditions providing an exponential decrease of the error with
respect to the number of terms. The results of the consistent use of tensor formats
can be regarded as a base for a new class of rank truncated iterative eigensolvers
with almost linear complexity in the univariate problem size that improves dramatically
the traditional methods of linear scaling in the volume size. Tensor methods
can be applied to solving large scale spectral problems in the computational quantum
chemistry, for example to the Schr¨odinger, Hartree-Fock and Kohn-Sham equations in
electronic structure calculations. The results of numerical experiments clearly indicate
the linear-logarithmic scaling of low-rank tensor method in the univariate problem size.
The algorithms work equally well for the computation of both, minimal and maximal
eigenvalues of the discrete elliptic operators.