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The problem of the determination of the values of a field on a surface from its values on a surface to which it has propagated is shown to have a unique solution if the field satisfies any linear elliptic partial differential equation. Suppose that a scalar field ~ is the solution of a linear second order elliptic partial differential equation L ~ =0, (1) in a domain D bounded by two closed surfaces S 1 and S 2, (see fig. I). The equation (1) can for instance be the Helmholtz equation (V 2 + k2n2)~b = 0, valid in a medium with variable index of refraction, or the time independent Schr6dinger equation in the presence of an electromagnetic field, characterized by the vector potential A and the scalar potential ¢

Year: 1979

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