标题： 关于雅可比矩阵的反散射问题，其谱在区间、有限个区间的集合或正长度的康托集上 显示英文标题

标题： On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length

摘要： 求解具有快速衰减势的离散Sturm-Liouville算子的反散射问题，可以得到反射系数$s_\pm$和可逆算子$I+H_{s_\pm}$，其中$ H_{s_\pm}$是与符号$s_\pm$相关的Hankel算子。 Marchenko-Fadeev定理（在连续情况下）和Guseinov定理（在离散情况下），保证了反散射问题解的唯一性。 在本文中，我们提出一个自然的问题——能否找到一个精确条件，保证反散射问题唯一可解，并且算子$I+H_{s_\pm}$是可逆的？ 能否断言唯一性蕴含可逆性或反之？ 此外，我们不仅关注衰减势的情况，还关注渐近几乎周期势的情况。 因此，我们在这里结合了Sturm-Liouville算子反问题的两个主要研究情况：具有（几乎）周期势的反问题和具有快速衰减势的反问题。 显示英文摘要

摘要： Solving inverse scattering problem for a discrete Sturm-Liouville operator with the fast decreasing potential one gets reflection coefficients $s_\pm$ and invertible operators $I+H_{s_\pm}$, where $ H_{s_\pm}$ is the Hankel operator related to the symbol $s_\pm$. The Marchenko-Fadeev theorem (in the continuous case) and the Guseinov theorem (in the discrete case), guarantees the uniqueness of solution of the inverse scattering problem. In this article we asks the following natural question --- can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators $I+H_{s_\pm}$ are invertible? Can one claim that uniqueness implies invertibility or vise versa? Moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials. So we merege here two mostly developed cases of inverse problem for Sturm-Liouville operators: the inverse problem with (almost) periodic potential and the inverse problem with the fast decreasing potential.