A NEW METHOD FOR CALCULATING BOUND-STATES OF A LATTICE ELECTRON

States of an electron bound by an attractive potential on a discrete lattice are usually obtained as eigenvectors in correspondence with eigenenergies solving an eigenvalue equation. The equation is, however, satisfied by all the eigenenergies of a Hamiltonian, irrespective of whether they give bound or unbound states. No systematic method for obtaining only bound states has been known. Perturbational expansion with respect to the attractive potential does not work since bound states disappear if the potential vanishes. It is shown that we can obtain an approximate pair of eigenenergy and eigenfunction of a bound state by applying a decoupling approximation to the potential. The procedure can be repeated by using the approximate eigenfunction thus obtained as a new decoupling function. We can reach the true pair of eigenenergy and eigenfunction systematically by repeating the procedure several times.