扩散方程系数的半逆问题

一般地,扩散方程的系数q(x)与p(x)是由两组谱或者一组谱及其标准常数唯一确定的.运用Hochstadt与Lieberman的方法证明了:(a)如果给定区间[π/2,π]上的p(x)及区间[0,π]上的q(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数p(x);(b)如果给定区间[π/2,π]上的q(x)及区间[0,π]上的p(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数q(x).

A Half-Inverse Problem for the Coefficients of a Diffusion Equation

Generally speaking, the coefficients $q(x)$ and $p(x)$ in a diffusion operator can be uniquely determined by two spectra or one spectrum and norming constants. In this paper, by using the Hochstadt and Lieberman's method, it is shown that (a) if $p(x)$ is prescribed on the interval $[frac{pi}{2},pi]$ and $q(x)$ is full given on $[0,pi]$, then a single spectrum suffices to determine $p(x)$ on $[0,frac{pi}{2}]$; (b) if $q(x)$ is prescribed on the interval $[frac{pi}{2},pi]$ and $p(x)$ is full given on $[0,pi]$, then a single spectrum suffices to determine $q(x)$ on $[0,frac{pi}{2}]$.