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We construct analytically solvable Hill equations with smooth potentials and smooth solutions using a method of Wu and Shih. We then use Floquet–Bloch theory to analyze the band structure and find the Floquet–Bloch fundamental system of solutions of the resulting differential equation. Three examples of constructed potentials and their corresponding solutions are worked out.

Exact analytical solutions of two Hill’s equations that have continuously differentiable coefficients are obtained in terms of the Floquet–Bloch fundamental system. New features of the band structures of those equations are reported and investigated.

We consider light scattering from a binary (two-layered) photonic crystal. When the index of refraction of the incident medium exceeds that of either layer, total internal reflection may occur, which is a type of phase transition. At band edges (similar to the case where the index of refraction of the incident medium is less than those of the layers), the two Bloch waves describing propagation become degenerate and a hybrid Floquet mode is required to understand the response. For angles of incidence exceeding the second critical angle (when total internal reflection occurs in both layers), the allowed bands disappear entirely.

Using binary and sinusoidal photonic crystals as examples, it is shown how to construct two independent Floquet–Bloch waves for TM-polarized light in a lossless one-dimensional periodic structure. Particular attention is given to those special situations where either one Bloch wave and a hybrid Floquet mode develop inside the structure (instead of two Bloch waves) or both developed Bloch waves become periodic functions.

Accurate analytic approximations are developed for the band gap boundaries and surface waves of a 1D photonic crystal, making use of the semiclassical theory recently developed by the authors. These analytic results provide useful insight on systematics of surface states.

We design an N-cell anti-reflection coating for electron transport through an arbitrary periodic semiconductor heterostructure. Stability conditions are derived which allow one to make an Nth order zero of the reflection amplitude at the design energy. Examples are given for up to N&equal;4 cells, showing that 95&percent; average transmissivity can be obtained.

An analytic solution is presented to the problem of designing an antireflection coating to be added to an arbitrary periodic superlattice, so as to create an optimal electron mini-band-pass filter.

We solve the singularity problem of the transfer matrix method for a two-dimensional electron wave guide consisting of a sequence of cells. The method is reformulated in terms of intrinsic Bloch waves of a device cell. The resulting method is proved to be both stable and accurate. The transfer matrix is formulated without divergence problems in the open-mode approximation. The formula relating the N-cell transmission probability with that of one cell in the single-mode approximation takes exactly the same form as in the one-dimensional case.

We determine the binding energies of the model triton for two partly nonlocal interactions consisting of a local potential with a strong repulsive core and a nonlocal separable interaction acting only inside the core region, which is responsible for the occurrence of a continuum bound state at very high energies in the total interaction. The ground state of the three-particle system does not collapse in this case as for purely nonlocal interactions. The occurrence of a continuum bound state is therefore only a necessary but not a sufficient condition for such an unphysical collapse in few particle systems.

The energy spectrum of the model triton is studied for a series of rank-two separable twobody interactions, each of which has a fixed deuteron wave function and a continuum bound state fixed at high energy. In each case the three particle system collapses, having an unphysical deeply bound ground state. With increasing two body repulsion, the collapsed state becomes less bound. If the continuum bound state moves to higher energies, the collapsed state becomes more bound. Our results show that for non-local two-body interactions which support a continuum bound state, or a resonance pole sufficiently close to the real axis, the three particle system is not a low energy system.

Data on the isotope shift of the nuclear charge densities of the tin isotopes and of their mean square charge radii are considered from the viewpoint of the Hartree-Fock and Droplet Models. Attention is paid to the electromagnetic corrections of order 1/m2. Through a leptodermous analysis of charge and matter densities we find: (i) differences of m.s. charge radii are mainly sensitive to the external region, (ii) the electromagnetic corrections are important in that region, (iii) despite different leptodermous properties, forces Ska and GO-P lead to the same neutron skin thickness. The average rate of variation of the proton m.s. radius is interpreted as favouring either a low valueQ˜30 MeV for the droplet model asymmetry parameter, or a zero value for the saturation asymmetry parameterL.

The isotopic charge density differences among the isotopes of lead, as determined by elastic electron scattering, contain a remarkable contribution from the electromagnetic spin-orbit (EMSO) interaction. We show that the size of the effect depends upon the effective interaction employed, and discuss how it may be exploited to determine the 3p1/2 state occupation in the ground state wave functions of the several lead isotopes.