Inverse Problem for the Schr\"odinger Equation with Non-self-adjoint Matrix Potential

We consider the dynamical system with boundary control for the vector Schr\"odinger equation on the interval with a non-self-adjoint matrix potential. For this system, we study the inverse problem of recovering the matrix potential from the dynamical Dirichlet--to--Neumann operator. We first provide a method to recover spectral data for an abstract system from dynamic data and apply it to the Schr\"odinger equation. We then develop a strategy for solving the inverse problem for the Schr\"odinger equation using this method with other techniques of the Boundary control method.