Moment problems for Jacobi matrices and inverse problems for systems of many coupled oscillators.

摘要

In this thesis we study a family of inverse problems for finite-dimensional Jacobi matrices. The family of problems arises from the undamped linearized dynamics for a system of many coupled oscillators, and is closely connected to the classical moment problem in analysis and the attendant theory of orthogonal polynomials.; In the physical context one considers a differential equation of the form x&d3;+Ax=M-1f, where the constant coefficient matrix A is tridiagonal, M is a diagonal matrix of mass terms, and f is an external force. For each fixed pair of indices ( i, j) let Gij( t) denote the displacement xj( t) of the jth component of the system in response to a unit impulse force f(t) = delta( t)ei applied to the ith component. We pose the following inverse problem: Given the function Gij, determine the matrices A and M. In general this problem is nonlinear, the solution is not unique, and the dimension of a solution is not determined or even bounded. This formulation of the family of inverse problems, corresponding to all possible values of i and j, is new and our work extends known results for the special case i =j = 1.; The above family of physical inverse problems can be reduced to the following matrix-theoretic statement. Fix a Jacobi matrix J and a pair of indices (i, j). Let mn denote the (i, j)-entry of the nth power Jn of J. Given the sequence mnn≥0 up to a positive scalar multiple, determine J. The latter problem may be viewed as a variant of the classical moment problem which dates back to the 19th century.; Our objective in this thesis is to describe the manifold of least-dimensional solutions to each inverse problem in the given family. For each specific problem in the family we define a pair of polynomials, the characteristic polynomial and the composite polynomial, which can be computed from the given data. We show that least-dimensional solutions to the problem correspond to divisors modulo the characteristic polynomial of the composite polynomial. Our description of the solution manifold depends on a detailed analysis of the location of the zeros of the composite polynomial. This analysis is facilitated by formulas we derive by means of the flip transpose. Ultimately, we give an explicit parameterization of the solution manifold over a connected semi-algebraic subset of the vector space of real univariate polynomials.; As a byproduct of this analysis, we obtain general methods to construct orthogonal polynomials, which has allowed us to show for the first time that well-known upper bounds on the number of common zeros between orthogonal polynomials are in fact attained.