The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
Mehdi Dehghan - One of the best experts on this subject based on the ideXlab platform.
-
the general Jacobi Matrix method for solving some nonlinear ordinary differential equations
Applied Mathematical Modelling, 2012Co-Authors: M R Eslahchi, Mehdi DehghanAbstract:Abstract In this paper, we obtain the approximate solutions for some nonlinear ordinary differential equations by using the general Jacobi Matrix method. Explicit formulae which express the Jacobi expansion coefficients for the powers of derivatives and moments of any differentiable function in terms of the original expansion coefficients of the function itself are given in the Matrix form. Three test problems are discussed to illustrate the efficiency of the proposed method.
Yu. M. Berezansky - One of the best experts on this subject based on the ideXlab platform.
-
Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices
Functional Analysis and Its Applications, 2020Co-Authors: Yu. M. Berezansky, A. A. Mokhon’koAbstract:The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution u(t), t ∈, [0, ∞), of the problem, one assigns a self-adjoint semi-infinite Jacobi Matrix J(t) whose spectral measure dπ(λ; t) undergoes simple evolution in time t. The solution of the Cauchy problem goes as follows. One writes out the spectral measure dπ(λ; 0) for the initial value u(0) of the solution and the corresponding Jacobi Matrix J(0) and then computes the time evolution dπ(λ; t) of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi Matrix J(t) from dπ(λ; t) and hence finds the desired solution u(t).
-
The Integration of Double-Infinite Toda Lattice by Means of Inverse Spectral Problem and Related Quetions
Methods of Functional Analysis and Topology, 2009Co-Authors: Yu. M. BerezanskyAbstract:The solution of the Cauchy problem for differential-difference double-infinite Toda lattice by means of inverse spectral problem for semi-infinite block Jacobi Matrix is given. Namely, we construct a simple linear system of three differential equations of first order whose solution gives the spectral Matrix measure of the aforementioned Jacobi Matrix. The solution of the Cauchy problem for the Toda lattice is given by the procedure of orthogonalization w.r.t. this spectral measure, i.e. by the solution of the inverse spectral problem for this Jacobi Matrix.
-
Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices
Functional Analysis and Its Applications, 2008Co-Authors: Yu. M. Berezansky, A. A. Mokhon’koAbstract:The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution u ( t ), t ∈, [0, ∞), of the problem, one assigns a self-adjoint semi-infinite Jacobi Matrix J ( t ) whose spectral measure dπ ( λ; t ) undergoes simple evolution in time t . The solution of the Cauchy problem goes as follows. One writes out the spectral measure dπ ( λ ; 0) for the initial value u (0) of the solution and the corresponding Jacobi Matrix J (0) and then computes the time evolution dπ ( λ; t ) of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi Matrix J ( t ) from dπ ( λ; t ) and hence finds the desired solution u ( t ). In the present paper, this approach is generalized to the case in which the role of J ( t ) is played by a block Jacobi Matrix generating a normal operator in the orthogonal sum of finite-dimensional spaces with spectral measure dπ (ζ; t ) defined on the complex plane. Some recent results on the spectral theory of these normal operators permit one to use the integration method described above for a rather wide class of differential-difference nonlinear equations replacing the Toda lattice.
Christian Remling - One of the best experts on this subject based on the ideXlab platform.
-
Uniqueness of reflectionless Jacobi matrices and the Denisov-Rakhmanov Theorem
Proceedings of the American Mathematical Society, 2010Co-Authors: Christian RemlingAbstract:If a Jacobi Matrix J is reectionless on ( 2; 2) and has a single an0 equal to 1, then J is the free Jacobi Matrix an 1, bn 0. I'll discuss this result and its generalization to arbitrary sets and present several applications, including the following: if a Jacobi Matrix has some portion of its an's close to 1, then one assumption in the Denisov-Rakhmanov Theorem can be dropped.
-
Uniqueness of reflectionless Jacobi matrices and the Denisov-Rakhmanov Theorem
arXiv: Spectral Theory, 2010Co-Authors: Christian RemlingAbstract:If a Jacobi Matrix $J$ is reflectionless on $(-2,2)$ and has a single $a_{n_0}$ equal to 1, then $J$ is the free Jacobi Matrix $a_n\equiv 1$, $b_n\equiv 0$. I'll discuss this result and its generalization to arbitrary sets and present several applications, including the following: if a Jacobi Matrix has some portion of its $a_n$'s close to 1, then one assumption in the Denisov-Rakhmanov Theorem can be dropped.
Z. Papp - One of the best experts on this subject based on the ideXlab platform.
-
Green’s Matrix from Jacobi-Matrix Hamiltonian
Journal of Mathematical Physics, 1997Co-Authors: B. Kónya, Géza Lévai, Z. PappAbstract:We propose two ways for determining the Green’s Matrix for problems admitting Hamiltonians that have infinite symmetric tridiagonal (i.e., Jacobi) Matrix form on some basis representation. In addition to the recurrence relation coming from the Jacobi-Matrix, the first approach also requires the Matrix elements of the Green’s operator between the first elements of the basis. In the second approach the recurrence relation is solved directly by continued fractions and the solution is continued analytically to the whole complex plane. Both approaches are illustrated with the non-trivial but calculable example of the D-dimensional Coulomb Green’s Matrix. We give the corresponding formulas for the D-dimensional harmonic oscillator as well.
-
green s Matrix from Jacobi Matrix hamiltonian
Journal of Mathematical Physics, 1997Co-Authors: B. Kónya, Géza Lévai, Z. PappAbstract:We propose two ways for determining the Green’s Matrix for problems admitting Hamiltonians that have infinite symmetric tridiagonal (i.e., Jacobi) Matrix form on some basis representation. In addition to the recurrence relation coming from the Jacobi-Matrix, the first approach also requires the Matrix elements of the Green’s operator between the first elements of the basis. In the second approach the recurrence relation is solved directly by continued fractions and the solution is continued analytically to the whole complex plane. Both approaches are illustrated with the non-trivial but calculable example of the D-dimensional Coulomb Green’s Matrix. We give the corresponding formulas for the D-dimensional harmonic oscillator as well.
Joao Da Providencia - One of the best experts on this subject based on the ideXlab platform.
-
an algorithm for constructing a pseudo Jacobi Matrix from given spectral data
Numerical Linear Algebra With Applications, 2013Co-Authors: Natalia Bebiano, Susana Furtado, Joao Da ProvidenciaAbstract:SUMMARY The main purpose of this paper is the extension of the classical spectral direct and inverse analysis of Jacobi matrices for the non-self-adjoint setting. Matrices of this class appear in the context of non-Hermitian quantum mechanics. The reconstruction of a pseudo-Jacobi Matrix from its spectrum and the spectra of two complementary principal matrices is investigated in the context of indefinite inner product spaces. An existence and uniqueness theorem is given, and a strikingly simple algorithm, based on the Euclidean division algorithm, to reconstruct the Matrix from the spectral data is presented. A result of Friedland and Melkman stating a necessary and sufficient condition for a real sequence to be the spectrum of a non-negative Jacobi Matrix is revisited and generalized. Namely, it is shown that a suitable set of prescribed eigenvalues defines a unique non-negative pseudo-Jacobi Matrix, which is J-Hermitian for a fixed J. Copyright © 2012 John Wiley & Sons, Ltd.
-
An algorithm for constructing a pseudo‐Jacobi Matrix from given spectral data
Numerical Linear Algebra With Applications, 2012Co-Authors: Natalia Bebiano, Susana Furtado, Joao Da ProvidenciaAbstract:SUMMARY The main purpose of this paper is the extension of the classical spectral direct and inverse analysis of Jacobi matrices for the non-self-adjoint setting. Matrices of this class appear in the context of non-Hermitian quantum mechanics. The reconstruction of a pseudo-Jacobi Matrix from its spectrum and the spectra of two complementary principal matrices is investigated in the context of indefinite inner product spaces. An existence and uniqueness theorem is given, and a strikingly simple algorithm, based on the Euclidean division algorithm, to reconstruct the Matrix from the spectral data is presented. A result of Friedland and Melkman stating a necessary and sufficient condition for a real sequence to be the spectrum of a non-negative Jacobi Matrix is revisited and generalized. Namely, it is shown that a suitable set of prescribed eigenvalues defines a unique non-negative pseudo-Jacobi Matrix, which is J-Hermitian for a fixed J. Copyright © 2012 John Wiley & Sons, Ltd.