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A systematic method from the discreteness to the continuity is presented for the dislocation equation of the triangular lattice. A modification of the Peierls equation has been derived strictly. The modified equation includes the higher order corrections of the discrete effect which are important for the core structure of dislocation. It is observed that the modified equation possesses a universal form which is model-independent except the factors. The factors, which depend on the detail of the model, are related to the derivatives of the kernel at its zero point in the wave-vector space. The results open a way to deal with the complicated models because what one needs to do is to investigate the behaviour near the zero point of the kernel in the wave-vector space instead of calculating the kernel completely.

The exchange coupling energy for two magnetized monolayers embedded symmetrically in a metal and polarized in an arbitrary direction has been investigated in contact interaction approximation. Since the model can be solved exactly in particular for free-electron case, the coupling energy contributed from both extended state electrons and bound state electrons is calculated rigorously. For weak interaction, it is found that the leading term in the power-series expansion of density of states can give a correct coupling energy compared with rigorous one while extended state electrons give a much larger coupling energy. Furthermore, the relevant problems such as phase shift, 90° coupling and lattice effects have been discussed; an asymptotic expression of the interlayer coupling has been derived in a different way and used to calculate the exchange energy between magnetic layers in copper with Fermi surface obtained from de Haas-van Alphen effect.

An asymptotic expression for the density oscillation induced by an interface has been derived analytically by using a method which is equivalent to stationary phase approximation. The effects of the geometric structure of the Fermi surface and reflected amplitude are shown explicitly. The results provide a model-independent interpretation for some common features observed recently in the magnetic multilayer experiments.

A transfer matrix which transfers the electron states from one side of the interface to the other is defined for crystalline multilayer with layers joined coherently. The total transfer matrix for several sequential interfaces can be simply obtained by multiplication of the matrices. The bound states as well as the extended states can be naturally described in terms of transfer matrix. As an illustration of using transfer matrix, the equation of quantum well states for sandwich structure have been given in this way.

For a monatomic one-dimensional chain of particles interacting via a nearest-neighbor purely anharmonic potential of any even order (such a chain is called purely anharmonic chain), it is proved that there really exist exact localized modes.