The boundary value problem for the Helmholtz equation in the m -dimensional free space is considered. The problem is reduced to the Lippmann–Schwinger integral equation over the inhomogeneity domain. The operator of the integral equation is shown to be an invertible Fredholm operator. The inverse coefficient problem is considered. An application of the two-step method reduces the inverse problem to the source-type integral equation with a smooth kernel. Special classes of solutions to this equation are introduced. The uniqueness of a solution to the integral equation of the first kind is proved in the defined function classes.

中文翻译：

---

ℝ m 中亥姆霍兹方程的直接和反标量散射问题

考虑了亥姆霍兹方程在m维自由空间中的边值问题。问题简化为不均匀域上的李普曼-施温格积分方程。积分方程的算子被证明是一个可逆的 Fredholm 算子。考虑反系数问题。两步法的应用将逆问题简化为具有平滑核的源型积分方程。介绍了该方程的特殊类别的解。在定义的函数类中证明了第一类积分方程解的唯一性。