On the semiclassical green's function in the energy-representation

We present a detailed discussion of the semiclassical approximation for the Green's function in the energy representation, G(qⁱ,q^j; E). In particular, we describe the way in which classical catastrophes influence the semiclassical approximation, and present a practical scheme to determine the phases with which the contributions due to the various classical trajectories enter the semiclassical result.     

A classical green's function approach to vibrational spectra of biopolymers

Classical green's function methods can be used to calculate the spectra of large molecules by essentially putting together the spectra of smaller parts. The methods are useful where the parts are connected by few valence bonds.     

Interpretation of the photoelectron spectrum of norbornadiene: A green's function approach

The ionization potentials of norbornadiene are calculated by an ab initio many-body Green's function method. The photoelectron spectrum as recorded by Heilbronner and co-workers can thus be interpreted. The first two ionization potentials are 5b₂ (π) and 7a₁ (π) (with increasing binding energy) in agreement with previous investigations.     

Electronic structure of the ideal silicon vacancy by the muffin-tin green's function method

The muffin-tin Green's function method is applied to the ideal vacancy in silicon in the extended-charge single site approximation. The characteristics of the partially occupied gap state are found to be strongly dependent on the extent of the charge perturbation outside of the vacancy muffin-tin.     

Infra-red asymptotic behaviour of the one-fermion green's function in a scalar model with isospin

In a theory where massive fermions interact with a massless scalar field of isospin 1, the behaviour of the one-fermion Green's function is found to differ from the free Green's function by a factor $\text{(1 ? (2g}{}^2\text{/π}{}^2\text{)1}\text{n}\text{m|x ? y|)}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>8</mtext>}}$, in the limit of large separation |x ? y|.     

Ti+2 impurity states in ZnSe by the green's function method

A generalization of the molecular multiple scattering method to potentials consisting of a muffin-tip potential plus a spherically symmetric potential is used in calculations of the electronic structure of titanium impurity in zinc selenide. The results are compared with that obtained by the usual multiple-scattering method and with available experimental data.     

Geometric dependence of casimir energy and scalar green's functions

It is well known that the Casimir Energy depends on the geometry of the conducting cavity. In this note, scalar Green's functions are used to determine the Casimir energy's dependence on terms bilinear in the extrinsic curvatures of the cavity's surface, and thus to resolve the controversy over the Casimir energy's finiteness.     

Binary collision expansion and partial green's functions of Kadanoff-Baym

A new formula for the binary collision expansion of the unitary operator U (t₂, t₁) is proposed. The formula is applied to the expansion of the partial Green's functions of Kadanoff-Baym in powers of the correct binary scattering amplitude. It is shown that certain linked diagrams of left-multidentate structure should be taken into account. The duration of the binary collision is seen to play an important role in the rigorous formulation. Upon neglecting this duration, a useful approximation is found for the analysis of correlations on a macroscopic time scale.     

A green's function analysis of atomic and molecular (e, 2e) reactions

Investigation of Atomic and molecular (e, 2e) spectra will be discussed in terms of a Green's function approach. The energy, intensity and momentum distribution of energy levels observed by electron coincidence ionization spectroscopy, are directly related to the poles, pole strengths and generalized overlap amplitude of the one particle propagator or Green's function. The theoretical calculation of these observable quantities via the Green's function technique will be discussed. In particular, the position and intensity of satellite (or “shakeup”) lines, relative to the main lines, will be analysed in some detail.     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

Hydrogenic atom multiphonon ionization by intense fields with green's method

An expression for n-photon cross-section ionization of hydrogenic atoms by intense linearly polarized light is obtained using the second quantification formalism for the radiation and the Green's function method. It is shown that it reduces into Stobbe's formula in the one photon case.     

The equations of motion for green's functions and superconductivity solutions in particle physics

Three quantum field models are treated in a unified manner by using the method familiar from statistical many-body theory of solving the equations of motion for the Green functions in a particle number representation, with the following results: While the Nambu model does undergo dynamical symmetry breaking in its (1 + 1) dimensional version, it fails to do so in the original (3 + 1) dimensional case. For spinor QED, we get a critical value of the coupling parameter $\text{α}{}_{\mathrm{<mtext>c</mtext>}}\text{≡}\text{e}{}_{\mathrm{<mtext>c</mtext>}}{}^2\text{4π}\text{= 2π}$, above which the model becomes unstable.     

Application of the Green's function method to lattice vibrations in thin films

The Green's function method is used to investigate the lattice vibrations in a simple cubic thin film with nearest neighbour interactions. Appropriate Green's functions for the study of the surface waves for (001) and (011) orientations of the surface are determined. For a thin film with two parallel (001) free surfaces the atomic force tensors at the surface are modified in order to satisfy the rotational invariance condition. The surface waves are determined and a detailed discussion is made for the long wave-length limit k_y = 0 and k_x ? 1. For a given k_x there is a critical thickness below which only an antisymmetrical solution can exist. For the (011) orientation only the case of a single free surface is discussed.     

Functional approach to the parametric derivatives of renormalized green's functions with multiple insertions

Functional derivation of formulae for parametric derivatives of renormalized Green's functions with multiple insertions of composite operators is presented. An application to the derivative with respect to the ultraviolet cutoff in the Pauli–Villars regularized ?⁴₄ theory is given.     

Some properties of green's functions in the non-linear Rξ gauge

In the non-linear R_ξ gauge, the W^? boson field satisfies the naive Ward identity and in consequence one-loop contributions give zero neutrino charge, which they do not in the conventional R_ξ gauge. Further, in this non-linear gauge, the separate diagrams contributing to the photon self-energy are transverse and their divergent parts gauge independent.     

Weyl's association,Wigner's function and affine geometry

According to Weyl one may associate a function with a dynamical operator; these functions depend on the parameters p and q and can be displayed in a p, q manifold, the W space. In the classical limit the W space becomes the phase space parametrised by the canonical variables. The function associated in this manner with the density operator is Wigner's function. It turns out that if Wigner's function is a delta function it cannot represent the density operator of a physically realisable state unless the argument of the delta-function is linear in the parameters a and q. In all other cases Wigner's function associated with a physically realisable state has a finite width, proportional to $\text{h}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Consequently straightness (linear combination of p and q) has a fundamental significance in the W space. Since this property is preserved under linear inhomogeneous transformations the W space will have a geometry generated by these transformations, the affine geometry of Euler, Moebius and Blaschke. In the present note we show how this comes about, how it simplifies the semiclassical approximations of Wigner's function, and makes one understand how in the classical limit this geometry is lost, allowing to be replaced by the geometry of canonical transformations.     

On Milne's quantum number function

The quantum number function N(E) introduced by Milne is studied in detail. It is shown that N(E) is not uniquely defined for energies different from the bound state values. The density of states $\text{d}\text{N}\text{d}\text{E}$ is nowhere unique and not necessarily positive.     

Green's function theory for the one-dimensional antiferromagnet with antisymmetric exchange

The spin wave spectrum for the one-dimensional antiferromagnet with antisymmetric exchange is obtained from second order Green's function theory. There is a noticeable asymmetry in the spin wave dispersion curve, which has an effect on the static correlation function and the correlation length.     

Basis sets for the study of point defects in solids using green's-function methods

A number of Green's-function methods have recently been used to study the electronic structure of point defects and impurities in semiconductors. These methods are based on the same underlying operator formalism. An important factor controlling the efficiency of any of these methods is the basis set used to represent the operators in matrix form. Here we point out that it is the physical character of the perturbation potential that should suggest the nature of suitable basis sets. For point defects, the perturbation potential has recently been demonstrated by self-consistent calculations to have a predominantly single-center character. A single-center basis set, therefore, appears to be more suitable than multicenter basis sets. We present formal arguments and calculations that indicate that a set of harmonic-oscillator eigenfunctions centered at the point-defect site is particularly suitable, largely because such functions concentrate their variational flexibility in a small volume.