Three theorems concerning diffraction by a strip or a slit

The problem of diffraction on a strip or a slit is under consideration. A functional equation of Wiener-Hopf type is derived; analytical restrictions are imposed on unknown functions. The solution of a functional problem (the spectrum of the scattered field) is proved to be a solution of an ordinary differential equation with rational coefficients known up to several numerical constants. A nonlinear differential equation describing the dependence of unknown constants on the width of the strip (slit) is derived. Connection between the methods developed here and the solution in the form of Schwarzschild series is established.