Analytical properties of theS-matrix applied to the resonance model of transition metals

Summary A method to construct ionic potentials for transition metals from the knowledge of the resonantd-band phase shiftη ₂=tg^-21/2W/(E-E_d) is developed. This is based on an inversescattering approach using the analytic properties of theS-matrix and Jost functions to reconstruct the interparticle potential in the spirit of the Marchenko and Gel'fand-Levitan formalisms. A specific form of the solution of the resulting integral equation is discussed and possible applications to a host of transition metal properties are outlined.

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Riassunto Si mette a punto un metodo per costruire potenziali ionici per metalli di transieione partendo dalla conoscenza dello sfasamento risonante della bandad,η ₂=tg^-21/2W/(E-E_d). Questo è basato su un approccio allo scattering inverso che tiene conto delle proprietà analitiche della matriceS e delle funzioni di Jost per ricostruire il potenziale tra particelle nell'ambito dei formalismi di Marchenko e Gel'fand. Si discute una forma specifica dell'equazione integrale risultante e si sottolineano le possibili applicazioni ad una grande quantità di proprietà dei metalli di transizione.

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Резюме Развивается метод для конструирования ионных потенциалов для переходных металлов, исходя из знания резонансного фазового сдвигаd-зоны,η ₂=tg^-21/2W/(E-E_d). Этот метод основан на обратной задаче рассеяния с использованием аналитических свойствS-матрицы и функций Йоста для конструирования междучастичного потенциала в духе формализма Марченко и Гельфанда-Левитана. Обсуждается специальная форма решения полученного интегрального уравнения. Указываются возможные применения для описания свойств переходных металлов.

     

Adiabatic curvature and theS-matrix

We study the relation of the adiabatic curvature associated to scattering states and the scattering matrix. We show that the curvature of the scattering states is not determined by the scattering data alone. However, for certain tight binding Hamiltonians, the Chern numbers are determined by theS-matrix and are given explicitly in terms of integrals of certain odd-dimensional forms constructed from the scattering data. Two examples, which are the natural scattering analogs of Berry's spin 1/2 magnetic Hamiltonian and its quadrupole generalization, serve to motivate the questions and to illustrate the results.Research supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship and Texas ARP Grant 003658-037Research supported in part by GIF, DFG and the Fund for Promotion of Research at the Technion     

Study of theS-matrix near bound states in the continuum

The behaviour ofS-matrix for potentials generating bound states in continuum in the neighbourhood of the positive bound state energies is studied. It is shown that unlike the case of usual negative energy bound states, theS-matrix does not have a pole at the positive bound state energy but becomes unity at the energy corresponding to bound states in continuum. Calculations ofS-waveS-matrix for a local potential constructed by Stillinger and Herrick and a separable nonlocal potential constructed by the present authors verify these results. Our results indicate that the bound states embedded in continuum constructedvia the von Neumann and Wigner procedure cannot be interpreted as resonances with zero width.     

The invariance of theS-matrix under point transformations in renormalized perturbation theory

We give a simple proof of the invariance of theS-matrix under point transformations of the fields in renormalized perturbation field theory.     

The statistical distribution of theS-matrix in the one-channel case

It is shown that the statistical distribution of an ensemble of one-channelS-matrices is uniquely determined by requiring that:1)S has poles only in the lower half of the energy plane and2) the functionS(E) is ergodic in a sense to be defined. A Monte Carlo calculation was performed to illustrate numerically the above statement.     

Plane-wave approximation in theS-matrix and the construction of bound-state wave functions

Three inversion problem approaches — byGelfand-Levitan, Marchenko andPetrá? [5] —in both non-relativistic and relativistic (Klein-Gordon) variants are used in an approximation scheme selected to construct bound-state wave functions which are advantageous for purposes of model hadron physics. This family of wave functions is created exclusively by theS-matrix quantities and derived in the approximation which requires the Jost function to be equal to the unity throughout the continuous spectrum (the plane-wave approximation). As a consequence of the difference in boundary conditions of the mentioned approaches, the resulting approximate wave functions are not identical, but it is shown that there exists a parallelism as to the form among them. This parallelism is explained more extensively in the non-relativistic case, where the transformation properties of alternative sets of functions are treated. In the present paper it is demonstrated that in the relativistic variants of the above approaches the non-relativistic plane-wave-approximation form of the constructed wave functions for a given bound state is preserved.     

Analytic structure in λ of theS-matrix for strongly singular potentials

The asymptotic behaviour in the -plane of solutions of the Schrödinger equation for scattering on singular potentials is investigated. The asymptotic behaviour of the Jost functions and theS-matrix is obtained. Furthermore, the general analytic form in the -plane of the Jost functions and theS-matrix is established. Some properties of the distribution of poles of theS-matrix are proved.On leave of absence from the Institute Ruder Bokovi, Zagreb, and the Zagreb University, Yugoslavia.     

Towards the construction of theS-matrix for the Liouville field theory

Using a Lie-theoretical approach an approximative nontrivialS-matrix for the (bosonic) Liouville field theory is constructed. OurS-matrix element is unitary and factorizable in terms of Gamma-functions. We comment also on previous papers on the same subject.     

Threshold singularities of theS-matrix and convergence of Haag-Ruelle approximations

We establish a one-to-one correspondence between the continuity properties of theS-matrix at the 2-particle threshold and the rate of convergence of the Haag-Ruelle approximations ψ(t) for asymptotic 2-particle states ψ with smooth wavefunctions. It turns out that the norm distance ∥ψ?ψ(t)∥ approaches 0 liktt ^?5/4 if theS-matrix has the normal threshold singularities and liket ^?3/4 in the exceptional case where the threshold has “absorbed” a bound state. These connections are valid both in relativistic quantum field theory and in non-relativistic models with short range interaction.     

Gauge dependence of world lines and invariance of theS-matrix in relativistic classical mechanics

The notion of world lines is studied in the constraint Hamiltonian formulation of relativistic point particle dynamics. The particle world lines are shown to depend in general (in the presence of interaction) on the choice of the equal-time hyperplane (the only exception being the elastic scattering of rigid balls). However, the relative motion of a two-particle system and the (classical)S-matrix are independent of this choice.A preliminary version of this paper was circulated as ICTP, Trieste, Internal Report IC/79/59On leave of absence from Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1184, Bulgaria     

Analytical properties of mean two-particle density matrix

The analytical properties of the mean two-particle density matrixZ(t)=d R ₁ d R ₂ <_c(R ₁:R ₂;t)(R ₁,R ₂;t)> in the right-hand halfplane of the complex variable t is considered; herec,v(R₁,R ₂; t) are single-particle density matrices. It is proven that, in the case of a Gaussian field, the function Z(t) is analytical in the region Ret > 0. It is shown that the frequency dependence of the light-absorption coefficient in disordered semiconductors is determined by the asymptote of the function Z(t) as t.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 46–50, June, 1987.     

Analytical properties of the mean field in many fermion systems

The importance of the dispersion relation linking the real and the imaginary parts of the mean field in many fermion systems is pointed out. General models are built for the imaginary part. The properties of the real part close to the Fermi level are discussed in relation with these models. Its analytical behaviour is determined by the analytical behaviour of the imaginary part in the same domain, but the numerical values can be strongly influenced by the large energy behaviour of the same quantity. The sensitivity upon the behaviour in the intermediate domain is described semiquantitatively. The nuclear matter case is briefly discussed. The generality of our results is pointed out.     

Analytical properties of the anisotropic cubic Ising model

We combine an exact functional relation, the inversion relation, with conventional high-temperature expansions to explore the analytic properties of the anisotropic Ising model on both the square and simple cubic lattice. In particular, we investigate the nature of the singularities that occur in partially resummed expansions of the partition function and of the susceptibility.     

Calculation of theS-matrix of the massive thirring model on a momentum space lattice

TheS-matrix of the massive Thirring model is calculated non-perturbatively using a momentum space discretization. This represents the first application of this technique to a fermionic model. We find agreement up to 3 digits with 2nd order perturbation theory in the small coupling regime. Renormalization effects are found to be small. Our results do not support the old analytic solution of theS-matrix by Berezin and Sushko, but do support that of Zamolodchikov and Zamolodchikov.     

On the regular holonomic character of theS-matrix and microlocal analysis of unitarity-type integrals

The previously proved results that every analytically renormalized Feynman integral is a regular holonomic function suggests that theS-matrix should be locally expressible as an infinite sum of regular holonomic functions. A regularity propertyR is formulated that expresses the condition that theS-matrix be locally expressible near each physical pointp as a convergent sum of regular holonomic functions, with each term enjoying some of the regularity properties of a corresponding Feynman integral. This propertyR holds at every physical pointp that has yet been analyzed by the methods of axiomatic field theory orS-matrix theory. Some analyticity properties of unitarity-type integrals are then examined under the assumption that theS-matrix satisfies propertyR and a weak integrability condition. These results rest heavily on some recently proved properties of regular holonomic functions.     

Analytical properties of conformal invariant four point functions

On the basis of the known group theoretical structure of the conformal invariant four point functions in the case of identical scalar fields ?(x) of scale dimension d,

$\text{<0|?(x}{}_1\text{)?(x}{}_2\text{)?(x}{}_3\text{)?(x}{}_4\text{)|0〉 = [(x}{}_1\text{?x}{}_4\text{)}{}^2\text{(x}{}_2\text{?x}{}_3\text{)}{}^2\text{]}{}^{\mathrm{-d}}\text{g(A,B),}$

the analytical properties of g(A, B) as a function of the harmonic ratios A and B are investigated. By imposing the conditions of spectrality and locality, and using invariance under complex dilatations, it is shown that the function g(A, B) must be homomorphic in the whole complex A-plane and B-plane with exception of the values A=0 and A=∞, B arbitrary, and B=0 and B=∞, A arbitrary.     

Analytical properties of the spectral characteristics of antireflection coatings

The spectral dependences of the coefficients of multilayer antireflection optical coatings are analyzed. The results can be used in research directed toward determining the lower limit of residual reflection.     

Analytical description of the electrodynamic properties of metallic photonic crystals

A method of approximate calculation of the interaction inverse matrix in the method of discrete dipoles is proposed. The knowledge of this matrix makes it possible to determine the optical response of a system to the action of an electromagnetic wave with an arbitrary shape, which can be represented as a combination of vector spherical wave functions. The number of calculation operations of the matrix in the proposed method is considerably smaller than in the case of its direct calculation. In the case of a change in the refractive index of scattering particles, two methods of approximate calculation of the interaction inverse matrix are also proposed. This makes it possible to calculate the optical response of systems with new characteristics without direct solving equations of a system with a large dimension. The accuracy of the methods is numerically determined for particles with spherical and cubic shapes. It is shown that the methods are computationally efficient and can be used to calculate the values of polarization vectors inside particles and extinction and absorption cross sections of systems.     

Analytical thermodynamical properties of a two-dimensional electron gas in a magnetic field

Analytical expressions for the thermodynamical properties of a two-dimensional electron gas in a perpendicular magnetic field are derived.This is accomplished by first deriving the general expression for the thermodynamical potential,and then employing this result to obtain the corresponding expression for the two-dimensional gas.The chemical potential and magnetization are studied as a function of temperature and magnetic field,and shown to be in agreement with prior work.It is also shown that the results are close to those obtained by assuming a Gaussian density of states for the Landau levels.