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This paper introduces and analyses the new grid-based tensor approach toapproximate solution of the elliptic eigenvalue problem for the 3Dlattice-structured systems. We consider the linearized Hartree-Fock equationover a spatial $L_1\times L_2\times L_3$ lattice for both periodic andnon-periodic problem setting, discretized in the localized Gaussian-typeorbitals basis. In the periodic case, the Galerkin system matrix obeys athree-level block-circulant structure that allows the FFT-baseddiagonalization, while for the finite extended systems in a box (Dirichletboundary conditions) we arrive at the perturbed block-Toeplitz representationproviding fast matrix-vector multiplication and low storage size. The proposedgrid-based tensor techniques manifest the twofold benefits: (a) the entries ofthe Fock matrix are computed by 1D operations using low-rank tensorsrepresented on a 3D grid, (b) in the periodic case the low-rank tensorstructure in the diagonal blocks of the Fock matrix in the Fourier spacereduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systemsin a box with Dirichlet boundary conditions are treated numerically by ourprevious tensor solver for single molecules, which makes possible calculationson rather large $L_1\times L_2\times L_3$ lattices due to reduced numericalcost for 3D problems. The numerical simulations for both box-type and periodic$L\times 1\times 1$ lattice chain in a 3D rectangular "tube" with $L$ up toseveral hundred confirm the theoretical complexity bounds for theblock-structured eigenvalue solvers in the limit of large $L$.

Block circulant and Toeplitz structures in the linearized Hartree-Fock equation on finite lattices: tensor approach