Reconstruction of pencils of differential operators on the half-line
The author deals with the inverse problem of reconstructing the coefficients of the differential pencil \[ ly\equiv y''+(\rho^2+i\rho q_1(x)+q_0(x))y=0,\quad x>0, \] and the linear form \[ U(y)=y'(0)+(i\rho h_1+h_0)y(0), \] where \(q_j \in W_1^j(0,\infty)\) are complex-valued functions and \(h_0, h_1\) are complex numbers, with \(h_1\neq \pm 1\). The Weyl function generated by the pair \(L=(l,U)\) is the main tool. New qualitative effects typical for pencils of differential operators are discovered. As a particular case, the inverse problem for the wave equation can be reduced to the inverse spectral problem for the above pencil. This allows to get a global solution to the inverse problem for the wave equation.
scientific article; zbMATH DE number 1517713
scientific article