An Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectra
An Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectra
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DOI：
10.1137/120896426
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发表时间：
2013-04
期刊：
影响因子：
--
通讯作者：
W. Rundell;P. Sacks
中科院分区：
文献类型：
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作者：
W. Rundell;P. Sacks
一个经典的反问题是“你能听到两端夹紧的弦的密度吗？”该数学模型产生了一个逆Sturm-Liouville问题的未知密度$\rho$，众所周知，答案是否定的：从夹紧端点条件的狄利克雷谱是不够的。有许多已知的方法可以添加额外的信息来获得肯定的答案，这些方法包括改变边界条件之一，重新计算频谱或给出每个本征模的能量--所谓的赋范常数。我们假设这两种变化都不可能。相反，我们将以一种我们可以规定和重新测量狄利克雷谱的方式，把已知的质量密度加到弦上。我们不能以最一般的形式回答唯一性问题，但会对应该选择什么样的“附加质量”以及这如何导致原始弦密度的重建给出一些见解。
A classical inverse problem is “can you hear the density of a string clamped at both ends?” The mathematical model gives rise to an inverse Sturm--Liouville problem for the unknown density $\rho$, and it is well known that the answer is negative: the Dirichlet spectrum from the clamped end-point conditions is insufficient. There are many known ways to add additional information to gain a positive answer, and these include changing one of the boundary conditions and recomputing the spectrum or giving the energy in each eigenmode---the so-called norming constants. We make the assumption that neither of these changes are possible. Instead we will add known mass-densities to the string in a way we can prescribe and remeasure the Dirichlet spectrum. We will not be able to answer the uniqueness question in its most general form, but will give some insight to what “added masses” should be chosen and how this can lead to a reconstruction of the original string density.