On the construction of a potential from Cauchy data
The authors consider the problem of determining the potential \(q(y)\in l^ \alpha(0,1)\) in the differential equation \(-\Delta u+ g(y)u= 0\), \((x,y)\in (0,1)\times (0,1)\); \(u(0,y)= u(1,y)= u(x,0)=0\), \(u(x,1)= f(x)\), \(u_ y(x,1)= g(x)\), where the condition \(f(0)= f(1)= 0\) is satisfied. The method of separation of variables is used to reduce the uniqueness question to the nonstandard inverse Sturm-Liouville problem. An asymptotic formula is developed that suggests that under appropriate conditions on the Cauchy pair \((f,g)\), \(q(y)\) is uniquely determined up to the mean. The authors give a numerical scheme for the recovery of \(q\). Numerical examples and the asymptotics show that reconstructing \(q\) from such data is similar to finding a function from its moments.
scientific article; zbMATH DE number 490551
scientific article