The levels of the two-particle Schrödinger operator corresponding to a crystal film

For a two-particle Schrödinger operator considered in a cell and having a potential periodic in four variables, we establish the existence of levels (i.e., eigenvalues or resonances) in the neighborhood of singular points of the unperturbed Green’s function and derive an asymptotic formula for these levels. We prove an existence and uniqueness theorem for the solution of the corresponding Lippmann-Schwinger equation.     

Operators with Singular Trace Conditions on a Manifold with Edges

We establish a new calculus of pseudodifferential operators on a manifold with smooth edges and study ellipticity with extra trace and potential conditions (as well as Green operators) at the edge. In contrast to the known scenario with conditions of that kind in integral form we admit in this paper ‘singular’ trace and Green operators. In contrast to standard conditions in the theory of elliptic boundary value problems (like Dirichlet or Neumann conditions) our singular trace conditions, in general, do not act on functions that are smooth up to the boundary, but admit a more general asymptotic structure. Their action is now associated with the Laurent coefficients of the meromorphic Mellin transforms of functions with respect to the half-axis variable, the distance to the edge.     

Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential

We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?₊ × S². This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd.     

Completeness theorem for singular differential pencils

A theorem completeness theorem of special vector functions induced by the products of the so-called Weyl solutions of a fourth-order differential equation and by their derivatives on the semiaxis is presented. We prove that such nonlinear combinations of Weyl solutions and their derivatives constitute a linear subspace of decreasing (at infinity) solutions of a linear singular differential system of Kamke type. We construct and study the Green function of the corresponding singular boundary-value problems on the semiaxis for operator pencils defining differential systems of Kamke type. The required completeness theorem is proved by using the analytic and asymptotic properties of the Green function, operator spectral theory methods, and analytic function theory.     

Singular value expansion for the Green function of Helmholtz operator

We consider the integral operator defined on a circular disk, and with kernel the Green function of the Helmholtz operator. We present an analytic framework for the explicit computation of the singular system of this kernel. In particular, the main formulas of this framework are given by a characteristic equation for the singular values and explicit expressions for the corresponding singular functions. We provide also a property of the singular values, that gives an important information for the numerical evaluation of the singular system. Finally, we present a simple numerical experiment, where the singular system computed by a simple implementation of these analytic formulas is compared with the singular system obtained by a discretization of the Green function of the Helmholtz operator.     

一类含多奇性项的Grushin型算子方程解的渐近性质

本文研究了一类含多个奇性项的Grushin型算子方程非平凡解的渐近性质问题.当方程的非线性项满足临界指数增长条件时,利用Moser迭代方法和分析技巧,获得了方程的非平凡解在奇点处的渐近性质,推广了Laplace算子的相关结果.     

A weakly supercritical mode in a branching random walk

The case of weakly supercritical branching random walks is considered. A theorem on asymptotic behavior of the eigenvalue of the operator defining the process is obtained for this case. Analogues of the theorems on asymptotic behavior of the Green function under large deviations of a branching random walk and asymptotic behavior of the spread front of population of particles are established for the case of a simple symmetric branching random walk over a many-dimensional lattice. The constants for these theorems are exactly determined in terms of parameters of walking and branching.     

Asymptotic Analysis of Localized Solutions to Some Linear and Nonlinear Biharmonic Eigenvalue Problems

In an arbitrary bounded 2‐D domain, a singular perturbation approach is developed to analyze the asymptotic behavior of several biharmonic linear and nonlinear eigenvalue problems for which the solution exhibits a concentration behavior either due to a hole in the domain, or as a result of a nonlinearity that is nonnegligible only in some localized region in the domain. The specific form for the biharmonic nonlinear eigenvalue problem is motivated by the study of the steady‐state deflection of one of the two surfaces in a Micro‐Electro‐Mechanical System capacitor. The linear eigenvalue problem that is considered is to calculate the spectrum of the biharmonic operator in a domain with an interior hole of asymptotically small radius. One key novel feature in the analysis of our singularly perturbed biharmonic problems, which is absent in related second‐order elliptic problems, is that a point constraint must be imposed on the leading order outer solution to asymptotically match inner and outer representations of the solution. Our asymptotic analysis also relies heavily on the use of logarithmic switchback terms, notorious in the study of Low Reynolds number fluid flow, and on detailed properties of the biharmonic Green’s function and its associated regular part near the singularity. For a few simple domains, full numerical solutions to the biharmonic problems are computed to verify the asymptotic results obtained from the analysis.     

Feynman operator calculus: The constructive theory

In this paper, we survey progress on the Feynman operator calculus and path integral. We first develop an operator version of the Henstock-Kurzweil integral, construct the operator calculus and extend the Hille-Yosida theory. This shows that our approach is a natural extension of operator theory to the time-ordered setting. As an application, we unify the theory of time-dependent parabolic and hyperbolic evolution equations. Our theory is then reformulated as a sum over paths, providing a completely rigorous foundation for the Feynman path integral. Using our disentanglement approach, we extend the Trotter-Kato theory.     

A characterization of the singular green operators in boutet de monvel's calculus via wedge sobolev spaces

The singular Green operators in Boutet de Monvel's calculus are characterized by the behavior of their iterated commutators with differentiations and vector fields tangential to the boundary on wedge Sobolev spaces.     

Fractional differential and integral operations via cumulative approach

In this study, a fractal operator model of cumulative processes is described. Accordingly, differential and integral operators of the fractional calculus are derived by the fractal operator model of a cumulative process. In order to exhibit the relation between our cumulative approach and fractional calculus, vertical motion of a body is handled within these frameworks. Thereby, regard to our assessments, the underlying physical mechanism of the success of the fractional differintegral operators in describing stochastic complex systems is uncovered to some extent.     

Spectral functions of singular differential operators with matricial coefficients

We study the asymptotic behavior of the Harish-Chandra function associated to a singular second order differential operator with matricial coefficients. The study is based on a detailed analysis of the asymptotic behavior of some eigenvectors of the operator from which results on the asymptotic behavior of the spectral function and the scattering matrix are derived.     

Bemerkung zu einer Arbeit von H. Schlosser

In [5], [7] is was shown that the operator of KLEIN-GORDON type is a spectral operator with critical points. In this paper we estimate where the critical points can be found. We are especially interested in finding sufficient conditions for the absence of singular critical points.     

On the Characterization of the Intermediate Space in Generalized Factorizations

The study of systems of singular integral equations of CAUCHY type, of TOEPLITZ and WIENER -HOPF operators leads to the question of existence and representation of generalized factorizations of matrix functions Φ in [L^P(Γ, σ)]^m. This yields a corresponding factorization of the basic multiplication or translation invariant operator A = A_CA₊, respectively, which can be seen as a splitting of a bounded into unbounded operators. The present paper is devoted to the study of the nature of the induced intermediate space Z = im A₊ =im A^?1, in particular, for Γ = ? and Φ ε ζ[C^β(?)]^{m × m} which is of special interest in certain applications. As we know, this implies detailed results about the structure and the explicit asymptotic behaviour of solutions of boundary and transmission problems near singular points with a relation also to eigenvalue problems which result from the classical series expansion approach or from the Mellin symbol calculus (see [13]).     

Asymptotic Toeplitz and Hankel operators on the Bergman space

In this paper the concept of asymptotic Toeplitz and asymptotic Hankel operators on the Bergman space are introduced and properties of these classes of operators are studied. The importance of this notion is that it associates with a class of operators a Toeplitz operator and with a class of operators a Hankel operator where the original operators are not even Toeplitz or Hankel. Thus it is possible to assign a symbol to an operator that is not Toeplitz or Hankel and hence a symbol calculus is obtained. Further a relation between Toeplitz operators and little Hankel operators on the Bergman space is established in some asymptotic sense.     

Oblique derivative and interface problems on polygonal domains and networks

We Investigate oblique derivative problems associated to the Laplace operator on a polygon and we extend our study to "polygonal interface problems" which are an extension to networks of the prevlous ones. We focus on the non variational character of such problems. We obtain index formulae, a calculus of the dimension of the kernel, an expansion of the 'semi-variational" (or weak) solutions into regular and singular parts and formulae for the coefficients of the singularities In such expanslons.     

Spectral and phase space analysis of the linearized non-cutoff Kac collision operator

The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmetric Boltzmann operator are also proven.     

Regions of positivity for polyharmonic Green functions in arbitrary domains

The Green function for the biharmonic operator on bounded domains with zero Dirichlet boundary conditions is in general not of fixed sign. However, by extending an idea of Z. Nehari, we are able to identify regions of positivity for Green functions of polyharmonic operators. In particular, the biharmonic Green function is considered in all space dimensions. As a consequence we see that the negative part of any such Green function is somehow small compared with the singular positive part.

     

Algebras with general commutation relations and their applications. II. Unitary-nonlinear operator equations

Various aspects of the calculus of functions of ordered self-adjoint operators are considered. Passage to the commutative limit in the case of general nonlinear commutation relations is studied. An asymptotic solution of the Cauchy problem and asymptotically self-similar solutions are constructed for unitary-nonlinear operator equations. Asymptotic solutions are found for the Hartree equation with Coulomb interaction.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 13, pp. 145–267, 1979.     

On the oblique derivative problem for second order elliptic equations

A calculus of polyhomogeneous paired Lagrangian distributions, associated to any two cleanly intersecting Lagrangain submanifolds, is constructed. The class is given an intrinsic characterisation using radial operators and a symbol calculus is developed. A class of pseudo—differential operators with singular symbols is developed within the calculus. This is used to give symbolic constructions of parametrices for operators of real principal type and paired Lagrangian distributions. The calculus is then applied to give a symbolic construction of the forward fundamental solution of the wave operator.