Stability of the Inverse Resonance Problem for Jacobi Operators

When the coefficients of a Jacobi operator are finitely supported perturbations of the 1 and 0 sequences, respectively, the left reflection coefficient is a rational function whose poles inside, respectively outside, the unit disk correspond to eigenvalues and resonances. By including the zeros of the reflection coefficient, we have a set of data that determines the Jacobi coefficients up to a translation as long as there is at most one half-bound state. We prove that the coefficients of two Jacobi operators are pointwise close assuming that the zeros and poles of their left reflection coefficients are ??-close in some disk centered at the origin.     

The Inverse Resonance Problem for Hermite Operators

In this paper the inverse resonance problem for the Hermite operator is investigated. The Hermite operator with the creation operator , the annihilation operator , and a finitely supported multiplication operator b, is an unbounded operator on ℓ ²(ℕ₀) having finitely many eigenvalues and infinitely many resonances (except for b=0, when there are no eigenvalues or resonances). It is shown that knowing the location of eigenvalues and resonances determines the potential b uniquely.      

奇型Sturm-Liouville算子的半逆问题

奇型Sturm-Liouville问题的势函数q（x）由两组谱唯一确定的,Sturm-Liouville算子的半逆问题讨论由一组谱和一半势函数唯一确定势函数q（x）.本文利用Koyunbakan和Panakhov的方法,证明一组谱和（π/2,π）上的势函数q（x）唯一确定（0,π）上Sturm-Liouville方程含有奇型的1sin2x的势函数q（x）.     

On the Inverse Resonance Problem

A new technique is presented which gives conditions under whichperturbations of certain base potentials are uniquely determinedfrom the location of eigenvalues and resonances in the contextof a Schrödinger operator on a half line. The method extendsto complex-valued potentials and certain potentials whose firstmoment is not integrable.     

The Inverse Spectral Problem for the Sturm-Liouville Operators with Discontinuous Coefficients

We study the inverse spectral problem for the Sturm–Liouville operator whose piecewise constant coefficient A(x) has discontinuity points x _k , k=1,...,n, and jumps A _k =A(x _k +0)/A(x _k -0). We show that if the discontinuity points x ₁,...,x _n are noncommensurable, i.e., none of their linear combinations with integer coefficients vanishes; then the spectral function of the operator determines all discontinuity points x _k and jumps A _k uniquely. We give an algorithm for finding x _k and A _k in finitely many steps.     

有限区间上积分-微分算子的逆结点问题

研究Dirichlet边界条件下的积分-微分算子逆结点问题.证明了积分-微分算子稠定的结点子集能够唯一确定[0,π]上的势函数q(x)和区域Do上的积分扰动核M(x-t)且给出了这个逆结点问题的解的重构算法.     

Inverse Jacobi multipliers

Inverse Jacobi multipliers are a natural generalization of inverse integrating factors ton-dimensional dynamical systems. In this paper, the corresponding theory is developed from its beginning in the formal methods of integration of ordinary differential equations and the “last multiplier” of K. G. Jacobi. We explore to what extent the nice properties of the vanishing set of inverse integrating factors are preserved in then -dimensional case. In particular, vanishing on limit cycles (in restricted sense) of an inverse Jacobi multiplier is proved by resorting to integral invariants. Extensions of known constructions of inverse integrating factors by means of power series, local Lie Groups and algebraic solutions are provided for inverse Jacobi multipliers as well as a suitable generalization of the concept to systems with discontinuous right-hand side.     

Inverse eigenvalue problems for Jacobi matrices

It is proved that a real symmetric tridiagonal matrix with positive codiagonal elements is uniquely determined by its eigenvalues and the eigenvalues of the largest leading principal submatrix, and can be constructed from these data. The matrix depends continuously on the data, but because of rounding errors the investigated algorithm might in practice break down for large matrices.