Abstract
This chapter focuses on Hamilton–Jacobi equation with an unbounded inhomogeneity. Physical considerations demand that there must be a scattering theory for the Coulomb potential, and so it was necessary to create a corresponding mathematical theory. This was done by Dollard for the Coulomb potential and was extended to more general long-range potentials. The chapter discusses the resulting theory of dressing transformations. It also discusses the scattering theory. It discusses the existence and properties of the wave operators W± for two self-adjoint operators H and H0 on a complex Hilbert space H. The chapter also discusses the construction of approximate solutions of the Hamilton–Jacobi equation.
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