An approximation problem (Q1180324)

The author proves that the set of scattering solutions to the Schrödinger equation is complete in the set of all solutions to this equation in \(L^ 2(D)\) where \(D\) is an arbitrary bounded domain in \(R^ 3\). The problem arises: how fast the coefficients of the linear combinations of the scattering solutions can grow when the accuracy of the approximation increases and a special solution to the Schrödinger equation is approximated? The special solution of special interest in inverse scattering theory is an exponentially growing at infinity solution. In the paper a method for solving the above problem is given and the estimates of the norm of the coefficients are derived. These results are very useful for the inverse scattering theory with fixed- energy data developed by the author.