On the Riesz basis property of the root functions of a discontinuous boundary problem

In this paper, we consider the nonself‐adjoint discontinuous Sturm Liouville operator with periodic (antiperiodic) boundary condition and compatibility conditions. Asymptotic formulas of eigenvalues and eigenfunctions of the operator are obtained. Using these accurate asymptotic formulas for eigenvalues and eigenfunctions, we prove the basisness of the root functions of the boundary value problem.     

On the nonself-adjoint ordinary differential operators with periodic boundary conditions

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the nonself-adjoint ordinary differential operator with periodic and antiperiodic boundary conditions, when coefficients are arbitrary summable complex-valued functions. Then using these asymptotic formulas, we obtain necessary and sufficient conditions on the coefficient for which the root functions of these operators form a Riesz basis.     

On the basis property of the root functions of differential operators with matrix coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.     

On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator

In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis inL ₂(0, 1).     

Spectral properties of an even-order differential operator

We present the spectral properties of an even-order differential operator whose domain is described by periodic and antiperiodic boundary conditions or the Dirichlet conditions. We derive an asymptotic formula for the eigenvalues, estimates for the deviations of spectral projections, and estimates for the equiconvergence rate of spectral decompositions. Our asymptotic formulas for eigenvalues refine well-known ones.     

The Ewald sums for singly,doubly and triply periodic electrostatic systems

When evaluating the electrostatic potential, periodic boundary conditions in one, two or three of the spatial dimensions are often required for different applications. The triply periodic Ewald summation formula is classical, and Ewald summation formulas for the other two cases have also been derived. In this paper, derivations of the Ewald sums in the doubly and singly periodic cases are presented in a uniform framework based on Fourier analysis, which also yields a natural starting point for FFT-based fast summation methods.     

On Bethe Vectors for the XXZ Model at Roots of Unity

We give a construction of creation operators responsible for appearance of Bethe vectors with the same eigenvalues of the transfer-matrix for the inhomogeneous arbitrary spin XXZ model at roots of unity with particular quasiperiodic boundary conditions. This construction generalizes a similar one, recently obtained by K. Fabricius and B. M. McCoy, for the homogeneous six-vertex model with periodic boundary conditions. Even in the latter case, the given proofs for the main formulas are simpler than the original one. Bibliography: 6 titles.     

Eigenfunction and Green’s Function Asymptotics for Hill’s Equation with Symmetric Single-Well Potential

Ukrainian Mathematical Journal - We determine the asymptotic formulas for eigenfunctions of Hill’s equation with symmetric single-well potential under periodic and semiperiodic boundary...     

Correlation Function of the Two-Dimensional Ising Model on a Finite Lattice: II

We calculate the pair correlation function and the magnetic susceptibility in the anisotropic Ising model on the lattice with one infinite and one finite dimension with periodic boundary conditions imposed along the second dimension. Using the exact expressions for lattice form factors, we propose formulas for arbitrary spin matrix elements, thus providing a possibility to calculate all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. We analyze passing to the scaling limit.     

Eigenvectors of the Baxter-Bazhanov-Stroganov τ<Superscript>(2)</Superscript>(<Emphasis Type="Italic">t</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>) model with fixed-spin boundary conditions

We give explicit formulas for the eigenvectors of the transfer matrix of the Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. We obtain these formulas from the formulas for the eigenvectors of the periodic BBS model by a limit procedure. The latter formulas were derived in the framework of Sklyanin’s method of separation of variables. In the case of fixed-spin boundaries, we solve the corresponding T-Q Baxter equations for the functions of separated variables explicitly. As a particular case, we obtain the eigenvectors of the Hamiltonian of the Ising-like ℤ_N quantum chain model. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 94–108, April, 2008.     

Mixed problem for a first-order partial differential equation with involution and periodic boundary conditions

The Fourier method is used to find a classical solution of the mixed problem for a first-order differential equation with involution and periodic boundary conditions. The application of the Fourier method is substantiated using refined asymptotic formulas obtained for the eigenvalues and eigenfunctions of the corresponding spectral problem. The Fourier series representing the formal solution is transformed using certain techniques, and the possibility of its term-by-term differentiation is proved. Minimal requirements are imposed on the initial data of the problem.     

Local bifurcations of plane running waves for the generalized cubic Schrödinger equation

For the generalized cubic Schrödinger equation, we consider a periodic boundary value problem in the case of n independent space variables. For this boundary value problem, there exists a countable set of plane running waves periodic with respect to the time variable. We analyze their stability and local bifurcations under the change of stability. We show that invariant tori of dimension 2, ..., n + 1 can bifurcate from each of them. We obtain asymptotic formulas for the solutions on invariant tori and stability conditions for bifurcating tori as well as parameter ranges in which, starting from n = 3, a subcritical (stiff) bifurcation of invariant tori is possible.     

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

一类拟线性系统的Lyapunov不等式

本文研究了一类拟线性系统，引入了反周期边值条件，基于反周期边值条件和数学分析的技巧，建立了新的Lyapunov不等式．     

Asymptotics for solutions of periodic difference systemsSupported in part by Fondecyt 1030535, 1080034, DGI-MATH UNAP 2008 and Diufro 120521.

Using dichotomies and periodic conditions, we obtain asymptotic formulas for solutions of a difference system of Poincaré type with periodic coefficients. Some results about the theory of existence of periodic solutions for linear difference systems are presented. At the end, an open problem on the asymptotic spectral representation is proposed.     

Duffing型方程组的边界值问题的解的存在性

给出了带Dirichlet边界条件、Neumann边界条件和周期边界条件的Duffing型方程组的两点边界值问题的解的几个存在性定理。     

Hadamard’s Formula Inside and Out

Our goal is to explore boundary variations of spectral problems from the calculus of moving surfaces point of view. Hadamard’s famous formula for simple Laplace eigenvalues under Dirichlet boundary conditions is generalized in a number of significant ways, including Neumann and mixed boundary conditions, multiple eigenvalues, and second order variations. Some of these formulas appear for the first time here. Furthermore, we present an analytical framework for deriving general formulas of the Hadamard type.     

Sliding and transversal motions on an inclined boundary in a periodically forced discontinuous system

In this paper, sliding and transversal motions on the boundary in the periodically driven, discontinuous dynamical system is investigated. The simple inclined straight line boundary in phase space is considered as a control law for such a dynamical system to switch. The normal vector field for a flow switching on the separation boundary is adopted to develop the analytical conditions, and the corresponding transversality conditions of a flow to the boundary are obtained. The conditions of sliding and grazing flows to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the corresponding local stability and bifurcation analysis of the periodic motion are carried out. Numerical illustrations of periodic motions with and without sliding on the boundary are given. The local stability analysis cannot provide the proper prediction of the sliding and grazing motions in discontinuous dynamical systems. Therefore, the normal vector fields of periodic flows are presented, and the normal vector fields on the switching boundary points give the analytical criteria for sliding and transversality of motions.     

Computation of the wave field in the problem of diffraction by a smooth periodic boundary

The results of computations of the wave field between the humps of a periodic boundary are presented. The results are established on the basis of the asymptotic formulas obtained by the authors in previous articles.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 67–72, 1989.     

Existence of solutions for sub-linear or super-linear operator equations

We investigate solutions to superlinear or sublinear operator equations and obtain some abstract existence results by minimax methods. These results apply to superlinear or sublinear Hamiltonian systems satisfying several boundary value conditions including Sturm-Liouville boundary value conditions and generalized periodic boundary value conditions, and yield some new theorems concerning existence of solutions or nontrivial solutions. In particular, some famous results about periodic solutions to superlinear or sublinear Hamiltonian systems by Rabinowitz or Benci and Rabinowitz are special cases of the theorems.