Semi-analytical solution for eigenvalue problems of lattice models with boundary conditions

Abstract

Closed-form relations for limiting eigenvalues of an infinite k-periodic spatial lattice in any number of dimensions d, and its semi-analytical extensions for any given size n of the lattice with free-free boundary conditions, are known. These are based on the eigenvalues of tridiagonal k-Toeplitz matrices (representing chains and d = 1), and their tensor products or sums. These semi-analytical methods for eigenvalues incur drastically lower computing costs than the direct numerical methods i.e. O(n) vs. O(n2) for the latter, and further they are more accurate for sufficiently large lattices approaching the limiting case (n > 100). This advantage in computing cost, accuracy, and numerical stability emerges as the original eigenvalue problem of nk in size is reduced to n eigen value problems each k in size, further making this approach very amenable to parallel computation when required. In this work, their errors in eigenvalues are compared with the errors of the direct numerical methods using special examples with high condition numbers. Secondly, in the absence of such analytical methods, one also resorts to periodic boundary conditions to limit the size of the numerical model representing a very large system. The convergence of numerical models with periodic boundary conditions to the limiting eigenvalues is highlighted, to emphasize the utility of the closed-form solution for the limiting eigenvalues. Thirdly, the fixed-fixed boundary conditions on a finite chain and their counterpart for periodic spatial lattices in higher dimensions (d > 1) are ad dressed using perturbations to tridiagonal k-Toeplitz matrices representing the first and last elements of the chain. Extensions of the semi-analytical methods for these cases by applying numerical methods only to update the few perturbed eigenvalues are proposed. An efficient extension for evaluating the eigenvectors in the case of real eigenvalues as required in most physical systems is also presented.