  The Half-Space Matching (HSM) method has recently been developed as a new
method for the solution of 2D scattering problems with complex backgrounds,
providing an alternative to Perfectly Matched Layers (PML) or other artificial
boundary conditions. Based on half-plane representations for the solution, the
scattering problem is rewritten as a system of integral equations in which the
unknowns are restrictions of the solution to the boundaries of a finite number
of overlapping half-planes contained in the domain: this integral equation
system is coupled to a standard finite element discretisation localised around
the scatterer. While satisfactory numerical results have been obtained for real
wavenumbers, wellposedness and equivalence to the original scattering problem
have been established only for complex wavenumbers. In the present paper, by
combining the HSM framework with a complex-scaling technique, we provide a new
formulation for real wavenumbers which is provably well-posed and has the
attraction for computation that the complex-scaled solutions of the integral
equation system decay exponentially at infinity. The analysis requires the
study of double-layer potential integral operators on intersecting infinite
lines, and their analytic continuations. The effectiveness of the method is
validated by preliminary numerical results.
