Two-time Green’s functions in superconductivity theory

Based on the method of the equations of motion for two-time Green’s functions, we derive superconductivity equations for different types of interactions related to the scattering of electrons on phonons and spin fluctuations or caused by strong Coulomb correlations in the Hubbard model. We derive an exact Dyson equation for the matrix Green’s function with the self-energy operator in the form of the multiparticle Green’s function. Calculating the self-energy operator in the approximation of noncrossing diagrams leads to a closed system of equations corresponding to the Migdal-Eliashberg strong-coupling theory. We propose a theory of high-temperature superconductivity due to kinematic interaction in the Hubbard model. We show that two pairing channels occur in systems with a strong Coulomb correlation: one due to the antiferromagnetic exchange in interband hopping and the other due to the coupling to spin and charge fluctuations in hopping within one Hubbard band. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 129–146, January, 2008.     

Solving the Charged-Particle Scattering Problem by Wave Packet Continuum Discretization

We generalize our previously developed packet continuum discretization method to take the long-range Coulomb repulsion in the charged-particle interaction into account. We derive an analytic finite-dimensional approximation for the exact Coulomb resolvent in the basis of stationary Coulomb wave packets. In the suggested approach, determining the so-called additional partial scattering phase shifts that appear because of the additional short-range interaction reduces to simple matrix algebra, and the related calculations can be performed using an arbitrary complete L₂ basis. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 393–410, December, 2005.     

“Asymptotically non-minkowskian” bubbles with gravitational repulsion

It is usually assumed that spherical domain walls can be described by the same equation of state as planar domains. In this case, the spherical domain walls should also demonstrate gravitational repulsion, which would contradict the Birkhoff theorem. However, this theorem does not apply to solutions that are not described by Minkowskian geometry at infinity. In the thin-wall formalism, we consider the solution of the Einstein equations that describes a spherical domain wall with gravitational repulsion and which is “asymptotically non-Minkowskian.” Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 346–352, November, 1997.     

The two-time Green’s function and the diagram technique

Based on constructing the equations of motion for the two-time Green’s functions, we discuss calculating the dynamical spin susceptibility and correlation functions in the Heisenberg model. Using a Mori-type projection, we derive an exact Dyson equation with the self-energy operator in the form of a multiparticle Green’s function. Calculating the self-energy operator in the mode-coupling approximation in the ferromagnetic phase, we reproduce the results of the temperature diagram technique, including the correct formula for low-temperature magnetization. We also consider calculating the spin fluctuation spectrum in the paramagnetic phase in the framework of the method of equations of motion for the relaxation function.     

Nonholonomic Mechanical Systems and Kaluza–Klein Theory

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.     

An asymptotic expression of the Schrödinger equation

The problem of solving the time–independent Schr?dinger equation for the motion of an electron of mass μ and charge –e (e > 0) in the field of two fixed Coulomb centers has been the subject of extensive studies in theoretical physics and quantum computation. In the present paper, after making a series of coordinate transformations, we apply the qualitative theory of nonlinear differential equations to the study of the Schr?dinger equation under certain parametric conditions, and obtain an asymptotic formula. The work has been presented at the International Conference on Quantum Computation and Quantum Technology, Texas A&M University, College Station, Texas, November 13-16, 2005. The author would like to thank the organizer Professor Goong Chen for his generous support. This work is also partly supported by UTPA Faculty Research Council Grant 119100.     

A (2+1)-dimensional fermion in the Coulomb and magnetic field backgrounds

In the problem of a two-dimensional hydrogen-like atom in a magnetic field background, we construct quasi-classical solutions and the energy spectrum of the Dirac equation in a strong Coulomb field and in a weak constant homogeneous magnetic field in 2+1 dimensions. We find some “exact” solutions of the Dirac and Pauli equations describing the “spinless” fermions in strong Coulomb fields and in homogeneous magnetic fields in 2+1 dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 105–118, April, 1999.     

The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes

This paper continues the theme of the recent work [Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal., to appear] and further develops the Petrov–Galerkin method for Fredholm integral equations of the second kind. Specifically, we study wavelet Petrov–Galerkin schemes based on discontinuous orthogonal multiwavelets and prove that the condition number of the coefficient matrix for the linear system obtained from the wavelet Petrov–Galerkin scheme is bounded. In addition, we propose a truncation strategy which forms a basis for fast wavelet algorithms and analyze the order of convergence and computational complexity of these algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Families of Okamoto–Painlevé pairs and Painlevé equations

In [as reported by Saito et al. (J. Algebraic Geom. 11:311–362, 2002)], generalized Okamoto–Painlevé pairs are introduced as a generalization of Okamoto’s space of initial conditions of Painlevé equations (cf. [Okamoto (Jpn. J. Math. 5:1–79, 1979)]) and we established a way to derive differential equations from generalized rational Okamoto–Painlevé pairs through deformation theory of nonsingular pairs. In this article, we apply the method to concrete families of generalized rational Okamoto–Painlevé pairs with given affine coordinate systems and for all eight types of such Okamoto–Painlvé pairs we write down Painlevé equations in the coordinate systems explicitly. Moreover, except for a few cases, Hamitonians associated to these Painlevé equations are also given in all coordinate charts. Mathematics Subject Classification (2000) 34M55, 32G05, 14J26     

Strong-coupling theory for multiband superconductors

We consider a microscopic theory of the strong coupling in multiband superconductors with an arbitrary electron-boson interaction. Based on the method of the equations of motion for two-time Green’s functions, we derive the Dyson equation with the self-energy operator in the form of the multiparticle Green’s function taking the interaction of electrons with phonons and spin fluctuations into account. We obtain a self-consistent system of equations for the normal and anomalous components of the Green’s function and the self-energy operator calculated in the approximation of noncrossing diagrams. We discuss the approximate solution of the system of equations taking only components of the self-energy operator that are diagonal with respect to the band index into account for studying superconductivity in iron-based compounds.     

Explicit Reciprocity Law for Lubin-Tate Formal Groups

In this article, using Fontaine＇s ФГ-module theory, we give a new proof of Coleman＇s explicit reciprocity law, which generalizes that of Artin-Hasse, Iwasawa and Wiles, by giving a complete formula for the norm residue symbol on Lubin-Tate groups. The method used here is different from the classical ones and can be used to study the Iwasawa theory of crystalline representations.     

Statistical field theory of a nonadditive system

Based on quantum field methods, we develop a statistical theory of complex systems with nonadditive potentials. Using the Martin-Siggia-Rose method, we find the effective system Lagrangian, from which we obtain evolution equations for the most probable values of the order parameter and its fluctuation amplitudes. We show that these equations are unchanged under deformations of the statistical distribution while the probabilities of realizing different phase trajectories depend essentially on the nonadditivity parameter. We find the generating functional of a nonadditive system and establish its relation to correlation functions; we introduce a pair of additive generating functionals whose expansion terms determine the set of multipoint Green’s functions and their self-energy parts. We find equations for the generating functional of a system having an internal symmetry and constraints. In the harmonic approximation framework, we determine the partition function and moments of the order parameter depending on the nonadditivity parameter. We develop a perturbation theory that allows calculating corrections of an arbitrary order to the indicated quantities.     

Long Range Scattering and Modified Wave Operators for the Maxwell–Schrödinger System II. The General Case

We study the theory of scattering for the Maxwell–Schr?dinger system in space dimension 3, in the Coulomb gauge. We prove the existence of modified wave operators for that system with no size restriction on the Schr?dinger and Maxwell asymptotic data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in partially solving the Maxwell equations for the potentials, substituting the result into the Schr?dinger equation, which then becomes both nonlinear and nonlocal in time. The Schr?dinger function is then parametrized in terms of an amplitude and a phase satisfying a suitable auxiliary system, and the Cauchy problem for that system, with prescribed asymptotic behaviour determined by the asymptotic data, is solved by an energy method, thereby leading to solutions of the original system with prescribed asymptotic behaviour in time. This paper is the generalization of a previous paper with the same title. However it is entirely self contained and can be read without any previous knowledge of the latter. Submitted: November 7, 2006. Accepted: November 14, 2006.     

Evaluation codes and plane valuations

We apply tools coming from singularity theory, as Hamburger–Noether expansions, and from valuation theory, as generating sequences, to explicitly describe order functions given by valuations of 2-dimensional function fields. We show that these order functions are simple when their ordered domains are isomorphic to the value semigroup algebra of the corresponding valuation. Otherwise, we provide parametric equations to compute them. In the first case, we construct, for each order function, families of error correcting codes which can be decodified by the Berlekamp–Massey–Sakata algorithm and we give bounds for their minimum distance depending on minimal sets of generators for the above value semigroup.     

The S-Matrix Structure and the Inverse Scattering Transform in the J-Matrix Approach

Using the analytic properties of the S-matrix, we obtain a system of inverse scattering transform equations for nonlocal potentials with Laguerre form factors. Coulomb repulsion can be present in the system.     

Nonlocal problems for the equations of motion of Kelvin-Voight fluids

For the equations of the motion of Kelvin-Voight fluids (0.1) and for the semilinear abstract differential Eqs. (0.11)–(0.17) arising in the theory of Sobolev equations (to which Eqs. (0.1) also belong), we study the four following nonlocal problems: 1) the solvability of the initial boundary problem for Eqs. (0.1) and the Cauchy problem for Eqs. (0.11)–(0.17) on the semiaxis 0<t≤∞; 2) the existence of periodic solutions of Eqs. (0.1) and Eqs. (0.11)–(0.17) with a free term periodic in t; 3) exponential stability theory for solutions of Eqs. (0.1) and Eqs. (0.11)–(0.17) as t→∞ and related problems; 4) attractor theory for Eqs. (0.1). Bibliography: 40 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 120–158, 1992.     

Hybrid Galerkin boundary elements: theory and implementation

Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we obtain a “hybrid” Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin method but a complexity more akin to that of a collocation or Nystr?m method. The method can be applied to a wide range of singular and weakly-singular first- and second-kind equations, including many for which the classical Nystr?m method is not even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular) meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory. Received January 22, 1998 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Influence of nucleus motion on the fine structure of a hydrogen-like atom with unequal particle masses

Using a modified quasipotential perturbation theory, we study the value of the fine shift of the energy S levels of hydrogen-like atoms due to the two-photon Coulomb interactions and the dependence of this value on the particle mass ratio parameter. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 325–338, December, 2006.     

The Bipolaron in the Strong Coupling Limit

The bipolaron are two electrons coupled to the elastic deformations of an ionic crystal. We study this system in the Fr?hlich approximation. If the Coulomb repulsion dominates, the lowest energy states are two well separated polarons. Otherwise the electrons form a bound pair. We prove the validity of the Pekar–Tomasevich energy functional in the strong coupling limit, yielding estimates on the coupling parameters for which the binding energy is strictly positive. Under the condition of a strictly positive binding energy we prove the existence of a ground state at fixed total momentum P, provided P is not too large. Tadahiro Miyao: This work was supported by Japan Society for the Promotion of Science (JSPS). Permanent address: The graduate school of natural science and technology, Okayama university, Okayama 700-8530, Japan. Submitted: December 14, 2006. Accepted: March 20, 2007.     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.