Download Topics in mathematical physics by Palamodov V. PDF
By Palamodov V.
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Example text
10) The points λ = 0, −1, −2, ... 11) where = ∂12 − ∂22 − ... − ∂n2 is the differential operator dual to the quadratic form q. In particular, the convolution φ →Z0 ∗ φ is the identity operator; this together with (10) means that the family of convolution operators {Zλ ∗} is a commutative group, which is isomorphic to the additive group of C. It is called the Riesz group. From (11) we see that k Zk = Z−k ∗ Zk = Z0 = δ0 dx This means that Zk is a fundamental solution for the hyperbolic operator k (which is not strictly hyperbolic for k > 1).
There are several types of such conditions. We suppose that the obstacle is impenetrable and the field u satisfies the Dirichlet condition u|∂K = 0. In this case the boundary ∂K is called also soft or pressure release surface in the context of the acoustic wave theory. In the case of Neumann condition ∂ν u|∂K = 0 it is called hard surface, the third condition appears for impedance surface. The total field u = ui + us is the sum of the incident plane wave and the scattered field us (θ; x) in X\K such that u satisfies the Dirichlet condition us |∂K = − exp (ık (θ, x)) |∂K and us fulfils the radiation condition.
This completes the proof. 7 Geometrical optics This is the ray method (Debay’s method) and similar methods for construction of high frequency approximations to solutions of the wave equation: auω = O ω −q where a is a wave operator (1) or a similar operator. One looks for an approximate solution of the form (WKB-form) uω (x, t) = exp(ıω(ϕ(x) + t))(a0 (x) + ω −1 a1 (x) + ... + ω −k ak (x)) = exp (ıωt) U (x, ω) 6 where the time frequency ω is a big parameter. Then the function U (x, ω) = exp(ıω(ϕ(x)))(a0 (x) + (ıω)−1 a1 (x) + ...