On the determination of differential equations with singularities and turning points
The paper deals with an inverse problem for stratified media, namely, one considers the system \[ \begin{cases} \frac{dy_1}{dx}=i\rho R(x)y_2,\;\frac{dy_2}{dx}=i\rho\frac{1}{R(x)}y_1,\quad x\in[0,T],&\\ y_1(0,\rho)=1,\;y_2(0,\rho)=-1,&\\ \end{cases} \tag{1} \] where \(T>0\), \(\rho\in{\mathbb C}\) is a spectral parameter and \(R\) is a real nonnegative function called the wave resistance. The function \(R\) is supposed to have the form \[ R(x)=\prod_{j=1}^N| x-\gamma_j|^{p_j-1}R_0(x), \] where \(0<\gamma_1<\cdots<\gamma_N<T\), \(p_j\in{\mathbb R}\), \(R_0,R_0',R_0^{\prime\prime} \in L^1[0,T]\), \(R_0>0\), \(R(0)=1\), \(R'(0)=0\). The inverse problem the paper deals with is: given the coefficient \[ r(\rho)=\frac{y_1(T,\rho)+R(T)y_2(T,\rho)}{y_2(T,\rho)-R(T)y_1(T,\rho)}, \] construct \(R(x)\). The authors show that \(R\) is uniquely determined by \(r\) and give a method of reconstruction of \(R\) from \(r\). The main idea is to associate to the above inverse problem an inverse problem for a Schrödinger operator on a half-axis, with a singular potential.
scientific article; zbMATH DE number 1997871