Analytic Renormalization

We develop a reliable parameter-free analytic continuation method for quantum\nmany-body calculations. Our method is based on a kernel grid, a causal spline,\na regularization using the second-derivative roughness penalty, and the L-curve\ncriterion. We also develop the L-curve averaged deviation to estimate the\nprecision of our analytic continuation. To deal with statistically obtained\ndata more efficiently, we further develop a bootstrap-averaged analytic\ncontinuation method. In the test using the exact imaginary-frequency Green's\nfunction with added statistical error, our method produces the spectral\nfunction that converges systematically to the exact one as the statistical\nerror decreases. As an application, we simulate the two-orbital Hubbard model\nfor various electron numbers with the dynamical-mean field theory in the\nimaginary time and obtain the real-frequency self-energy with our analytic\ncontinuation method, clearly identifying a non-Fermi liquid behavior as the\nelectron number approaches the half filling from the quarter filling. Our\nanalytic continuation can be used widely and it will facilitate drawing clear\nconclusions from imaginary-time quantum many-body calculations.\n

ABSTRACTThe stress field induced by an edge dislocation or a point force located near a coated triangle-like hole in an infinite plate is provided in this paper. Based on the method of analytical continuation and the technique of conformal mapping in conjunction with the alternation technique, a series solution for the displacement and stresses in the coating layer and the matrix is obtained analytically. Examples for the interaction between an edge dislocation and a coated triangle-like hole for various material constant combinations, coating thicknesses and shape factors are discussed. The analysis discovers that the so-called trapping mechanism of dislocations is more likely to exist near a coated triangle-like hole. The result shows that the dislocation will first be repelled by the coating layer and then attracted by a hole when the coating layer is slightly stiffer than the matrix. However, when the coating layer is sufficiently thin, the dislocation will always be attracted by a hole even the coating layer is stiffer than the matrix.

A new approach to get relations between the gluon condensates is shown to be utilization of the local Quantum Chromodynamics (QCD) sum rule via method of analytic continuation by duality in cooperation with the QCD low-energy theorem.

The laminar boundary layer in axial flow along a long thin cylinder with constant temperature is investigated by expanding in powers of a variable ξ that was used by Seban & Bond [J. Aero. Sci., 18 (1951), 671.]. The series solutions for the skin friction, the local Nusselt number and the other boundary-layer parameters are obtained up to about 20 terms, and by utilizing Pade approximants, their singular points on a complex plane are evaluated. Euler transformations then yield more convergent expansions, and by reflecting the known asymptotic behavior of the solutions on the extended series solutions, reasonable solutions are obtained for sufficiently large ξ. It is clarified that the solutions by the method of analytic continuation using Pade approximants are also reasonable. Pade approximants are applied further to velocity and temperature distributions.

Spherical functions on a real semisimple Lie group. A method of reduction to the complex case

We consider an Eshelby’s inclusion of arbitrary shape with prescribed uniform mid-plane eigenstrains and eigencurvatures in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite Kirchhoff laminated anisotropic thin plates. The inclusion has the same extensional, coupling and bending stiffnesses as the surrounding material. The boundary of the semi-infinite plate can be described by free, rigidly clamped and simply supported edges. We derive solutions of simple form by using the new Stroh octet formalism for the coupled stretching and bending deformations of anisotropic thin plates and the method of analytic continuation. In particular, real solutions of the far-field elastic fields induced by an inclusion of arbitrary shape are obtained. Specific examples of an elliptical inclusion in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite anisotropic plates are presented to demonstrate the obtained general solutions.

Using a new method of analytic continuation and the knowledge of parameter integration, anomalous imaginary part is written explicitly with the absorptive part (concerning t) of matrix element which connects initial or final state and intermediate one. The meaning of intermediate state is more clarified. In the case of π-meson deuteron scattering, all the singularities are determined. It is confirmed that these singularities agree with those obtained by performing the direct parameter integration of fourth order diagram.

Elastic study on singularities interacting with interfaces using alternating technique

The series representation of the lattice Green's function for the simple cubic lattice I(a)=π−3∫oπ∫oπ∫oπD−1dxdydz, where D=a-iε-cosx-cosy-cosz, around the singularity a=1 is obtained in fractional powers of a2−1 (convergent for |a2−1|<1), by the method of analytic continuation using a Mellin-Barnes type integral and also by use of the analytic continuation of 3F2 (, , ; , ; 1) as a function of the parameter. It gives leading and full expansions near the singularity a=1.

A general solution for the problems of rigid stamp indentation on a curvilinear hole boundary of an anisotropic elastic body is obtained by employing the Stroh formalism, the method of analytical continuation, and the technique of mapping a closed curve to a unit circle. With this general solution, two typical curvilinear holes are studied. One is an ellipse, the other is a polygon. Since the transformation functions used in our solutions are not always one to one, some of the solutions are not exact but only approximate. For example, the solutions to the problems of anisotropic plates containing an elliptic hole and isotropic plates containing a polygonal hole are exact, but the solutions to the problems of anisotropic plates containing a polygonal hole are only approximate. Because the solutions presented in this paper are new and no other analytical solution has been found in the literature, the correctness of the present results can only be checked analytically by their reduced forms such as those for isotropic media or those for the stress boundary value problems. To show the generality of our solutions and to see clearly the physical behavior of the indentation problems two numerical examples are given, and their related hoop stress and stress contour are also plotted.

The traditional theory of superconductivity relies on electron-phonon interaction. Explicitly, this was introduced into the theory of superconductivity by Eliashberg on the basis of Frohlich’s Hamiltonian and Migdal’s theorem . We will use Eliashberg’s model in the weak-coupling limit to derive kinetic equations, where both electrons, pairs and phonons are out of thermal equilibrium. Two methods are equally applicable for this task: the method of analytical continuation and Keldysh’s technique. We will demonstrate both methods in this Chapter. in this Chapter. The final kinetic equations will be expressed via the energy-integrated Green’s functions, which in the case of equilibrium problems are called Eilenberger functions. This corresponds to a quasiclassical approximation in nonequilibrium superconductivity of superconductors that have finite gaps. The interaction of Cooper pairs with the electrons and phonons plays an important role in the action of external fields, determining both the behavior of the order parameter and the nonequilibrium effects in the electron-phonon system.

A general analytical solution for the plane problem of elasticity about a reinforced elliptic hole embedded in an infinite matrix subjected to a point load is derived in this paper. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions for the displacement and stresses in the reinforcement layer and the matrix are derived explicitly in a series form. Some numerical results are provided to investigate the effects of the material combinations and geometric configurations on the interfacial stresses induced by a point force in this work.

A general solution to the thermoelastic problem of a circular inhomogeneity in an infinite matrix is provided. The thermal loadings considered in this note include a point heat source located either in the matrix or in the inclusion and a uniform heat flow applied at infinity. The proposed analysis is based upon the use of Laurent series expansion of the corresponding complex potentials and the method of analytical continuation. The general expressions of the temperature and stress functions are derived explicitly in both the inclusion and the surrounding matrix. Comparison is made with some special cases such as a circular hole under remote uniform heat flow and a circular disk under a point heat source, which shows that the results presented here are exact and general.

Central configurations with many small masses

We present an algorithm which, based on certain properties of analytic dependence, constructs boundary perturbation expansions of arbitrary order for eigenfunctions of elliptic PDEs. The resulting Taylor series can be evaluated far outside their radii of convergence—by means of appropriate methods of analytic continuation in the domain of complex perturbation parameters. A difficulty associated with calculation of the Taylor coefficients becomes apparent as one considers the issues raised by multiplicity: domain perturbations may remove existing multiple eigenvalues and criteria must therefore be provided to obtain Taylor series expansions for all branches stemming from a given multiple point. The derivation of our algorithm depends on certain properties of joint analyticity (with respect to spatial variables and perturbations) which had not been established before this work. While our proofs, constructions and numerical examples are given for eigenvalue problems for the Laplacian operator in the plane, other elliptic operators can be treated similarly.