Nonlinear spectral problem for a self-adjoint vector differential equation

We consider a spectral problem that is nonlinear in the spectral parameter for a self-adjoint vector differential equation of order 2n. The boundary conditions depend on the spectral parameter and are self-adjoint as well. Under some conditions of monotonicity of the input data with respect to the spectral parameter, we present a method for counting the eigenvalues of the problem in a given interval. If the boundary conditions are independent of the spectral parameter, then we define the notion of number of an eigenvalue and give a method for computing this number as well as the set of numbers of all eigenvalues in a given interval. For an equation considered on an unbounded interval, under some additional assumptions, we present a method for approximating the original singular problem by a problem on a finite interval.     

Solvability of the mixed initial boundary-value problem for a parabolic equation with a nonlocal condition of first kind

The spectral method is applied to solve the mixed initial boundary-value problem for a parabolic equation with nonhomogeneous boundary conditions, one of which is nonlocal. We prove existence and uniqueness of the generalized solution of this problem in the Sobolev class W ₂ ^1,0 and represent it as a biorthogonal series. We also consider optimal control by the right-hand side of the equation, which is constructed as a biorthogonal series in the root functions of the spectral problem.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 209–220, 2002.     

Reconstructing parameters of a medium from incomplete spectral information

The problem of the synthesis of a stratified medium with specified amplitude and phase properties is investigated. The wave propagation in the medium is described by a system of differential equations. The synthesis problem considered in the paper relates to inverse problems of spectral analysis with incomplete spectral information. Using the contour integral method we study properties of spectral characteristics and obtain algorithms for the solution of the synthesis problem for differential equations with singularities.     

Electromagnetic scattering by a smooth convex impedance cone

The problem of the diffraction of an electromagnetic planewave by a convex cone of arbitrary smooth cross-section withimpedance (Leontovich) boundary conditions is studied. The vectorproblem is reduced to that for the Debye potentials. By meansof Kontorovich–Lebedev integrals, two spectral functionsare introduced and the corresponding boundary value problemis formulated. The spectral functions for the potentials arefound to satisfy the Helmholtz equations on the unit sphereand to be coupled through non-traditional boundary conditionsof the impedance type with shifts on the spectral variable.The use of the Green theorem permits us to establish an integralformulation of the boundary value problem for the spectral functions.The formal asymptotic solution of the problem is then givenfor the case of a narrow cone. For this, two different methodsare given: a method of perturbation applied to the spectralintegral equations and an adaptation of the method of matchingthe asymptotic series in spectral domain. Both methods leadto the same closed-form result for the leading term of the scatteringdiagram asymptotics.     

Problem of small and normal oscillations of a rotating elastic body filled with an ideal barotropic liquid

An evolutionary problem of small motions of an ideal barotropic liquid filling a rotating isotropic elastic body is studied in the paper. Moreover, the corresponding spectral problem arising in the study of normal motions of the mentioned system is considered. First, we state the evolutionary problem, then we pass to a second-ordered differential equation in some Hilbert space. Based on this equation, we prove the uniqueness theorem for the strong solvability of the corresponding mixed problem. The spectral problem is studied in the second part of the paper. A quadratic spectral sheaf corresponding to the spectral problem was derived and studied. Problems of localization, discreteness, and asymptotic form of the spectrum are considered for this sheaf. The statement of double completeness with a defect for a system of eigenelements and adjoint elements and the statement of essential spectrum of the problem are proved.     

On a spectral condition for infrared singular quantum fields

The problem of formulating the spectral condition for vacuum expectation values of quantum fields with singular infrared behavior is discussed. It is shown that this problem is closely connected with the problem of extending the Paley-Wiener-Schwartz theorem to wider distribution classes. Studying this connection leads to a generalized spectral condition applicable to fields of arbitrarily high singularity.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 405–411, December, 1995.     

A Transmission Problem Involving a Parameter and a Weight

In the course of developing a spectral theory for non-selfadjoint elliptic boundary problems involving an indefinite weight function, there arises a transmission problem which has not hitherto been dealt with in a L_p setting. By educing our problem to one for ordinary differential equations with the aid of the Fourier transformation, we are able to resolve the transmission problem,that is to say, we are able to establish L_p estimates for its solutions which are supported in a neighbourhood of the origin, and this is p ecisely what is required for the furthe development of the spectral theory for the boundary problems cited above.     

On the uniform convergence in <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> of Fourier series for a spectral problem with squared spectral parameter in a boundary condition

We study the uniform convergence in C ¹ of the Fourier series of a Hölder function in a system of eigenfunctions corresponding to a spectral problem with squared spectral parameter in a boundary condition. We preliminarily study one more spectral problem with a spectral parameter in a boundary condition.     

On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval

An inverse spectral problem is studied for a non-selfadjoint Sturm-Liouville operator on a finite interval with an arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case. The connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved. A constructive procedure for solving the inverse problem is given.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

Stability estimate for a multidimensional inverse spectral problem with partial spectral data

In Bellassoued, Choulli and Yamamoto (2009) [4] we proved a log-log type stability estimate for a multidimensional inverse spectral problem with partial spectral data for a Schrödinger operator, provided that the potential is known in a small neighbourhood of the boundary of the domain. In the present paper we discuss the same inverse problem. We show a log type stability estimate under an additional condition on potentials in terms of their X-ray transform. In proving our result, we follow the same method as in Alessandrini and Sylvester (1990) [1] and Bellassoued, Choulli and Yamamoto (2009) [4]. That is we relate the stability estimate for our inverse spectral problem to a stability estimate for an inverse problem consisting in the determination of the potential in a wave equation from a local Dirichlet to Neumann map (DN map in short).     

Explicit solution and Darboux transformation for a new discrete integrable soliton hierarchy with 4×4 Lax pairs

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.     

超AKNS方程新的可积分解

该文介绍从3×3矩阵形式超谱问题出发, 构造新高阶矩阵形式超谱问题的方法.以超AKNS方程为例, 作者构造了5×5矩阵形式的超AKNS谱问题并且运用双非线性化方法,给出了超AKNS方程的新约束, 得到该约束下超AKNS方程新的可积分解.     

Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface

In one space dimension we address the homogenization of the spectral problem for a singularly perturbed diffusion equation in a periodic medium. Denoting by ε the period, the diffusion coefficient is scaled as ε². The domain is made of two purely periodic media separated by an interface. Depending on the connection between the two cell spectral equations, three different situations arise when ε goes to zero. First, there is a global homogenized problem as in the case without an interface. Second, the limit is made of two homogenized problems with a Dirichlet boundary condition on the interface. Third, there is an exponential localization near the interface of the first eigenfunction. Received: January 10, 2001; in final form: July 9, 2001?Published online: June 11, 2002     

Homogenization in general periodically perforated domains by a spectral approach

In this article we study the asymptotic behaviour as tends to 0 of the Neumann problem $-\Delta u_\epsilon+u_\epsilon=\epsilon$-periodic bounded open set of . The period cell of is equal to where is a regular open subset of the d-dimensional torus. We prove that if there exists a smallest integer such that the n-th non-zero eigenvalue of the spectral problem in satisfies , the limiting problem is a linear system of second order p.d.e.'s, of size n. By this spectral approach we extend in the periodic framework a result due to Khruslov without making strong geometrical assumptions on the perforated domain . Received: 20 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Integrable coupling hierarchy and Hamiltonian structure for a matrix spectral problem with arbitrary-order

We presented an integrable coupling hierarchy of a matrix spectral problem with arbitrary order zero matrix r by using semi-direct sums of matrix Lie algebra. The Hamiltonian structure of the resulting integrable couplings hierarchy is established by means of the component trace identities. As an example, when r is 2 × 2 zero matrix specially, the integrable coupling hierarchy and its Hamiltonian structure of the matrix spectral problem are computed.     

Numerical analysis of the dispersion of electroelastic waves in a cylinder

The wave problem of electroelastic waves in a cylinder is reduced to a spectral problem for a system of eight ordinary differential equations. We give an algorithm for numerical solution based on reducing the original boundary-value problem to four Cauchy problems and determining the roots of the dispersion equation by the method of bisection. One figure. Bibliogrpahy: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 149–153.     

Inversion of Trace Formulas for a Sturm-Liouville Operator

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.     

Uniform Convergence of Spectral Expansions in the Terms of Root Functions of a Spectral Problem for the Equation of a Vibrating Beam

In this paper we consider a spectral problem which describes bending vibrations of a homogeneous rod, in cross-sections of which the longitudinal force acts, the left end of which is fixed rigidly and on the right end is concentrated an elastically fixed load. We study the uniform convergence of spectral expansions in terms of root functions of this problem.     

Foundations of a Multi-way Spectral Clustering Framework for Hybrid Linear Modeling

The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curvature Clustering (TSCC) algorithm for solving the HLM problem and provides careful analysis to justify it. The TSCC algorithm is practically a combination of Govindu’s multi-way spectral clustering framework (CVPR 2005) and Ng et al.’s spectral clustering algorithm (NIPS 2001). The main result of this paper states that if the given data is sampled from a mixture of distributions concentrated around affine subspaces, then with high sampling probability the TSCC algorithm segments well the different underlying clusters. The goodness of clustering depends on the within-cluster errors, the between-clusters interaction, and a tuning parameter applied by TSCC. The proof also provides new insights for the analysis of Ng et al. (NIPS 2001). This work was supported by NSF grant #0612608.