Eigenfunction expansions and scattering theory for the wave equation in an exterior region
Eigenfunction expansions and scattering theory for the wave equation in an exterior region
复制标题
外部区域波动方程的本征函数展开和散射理论
DOI：
10.1007/bf00266571
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发表时间：
1966
影响因子：
2.5
通讯作者：
N. Shenk
中科院分区：
文献类型：
--
作者：
N. Shenk
Further properties of the outgoing and incoming subspaces enabled them to describe the spectral representations of {Ul (t)} explicitly as (generalized) eigenfunction expansions, from which they derived an explicit formula for the scattering operator.In this paper we present a different derivation of the eigenfunction expansions for {Ul (t)} and of the existence and representation of the associated scattering operator 6a~. This discussion is independent of the space dimension n> 2 and therefore extends the above results to the case of even space dimension. In addition, we present the parallel discussion for the group {0" 2 (t)} associated with the wave equation and zero Robin boundary condition in f2. We obtain the eigenfunction expansions for {Ul (t)} and {U2 (t)}, not as realizations of outgoing and incoming spectral representations, but through a separation of variables, modelled after the use of the Fourier transform in [1] in the construction of an eigenfunction expansion for the free-space wave equation. This construction, valid for arbitrary n, was based on the fact that the Fourier transform provides an eigenfunction expansion for the self-adjoint operator-A on L2 (R,), the plane waves {e-~ X'e: r(generalized eigenfunctions of the operator-A) giving the expansion.
Further properties of the outgoing and incoming subspaces enabled them to describe the spectral representations of {Ul (t)} explicitly as (generalized) eigenfunction expansions, from which they derived an explicit formula for the scattering operator.In this paper we present a different derivation of the eigenfunction expansions for {Ul (t)} and of the existence and representation of the associated scattering operator 6a~. This discussion is independent of the space dimension n> 2 and therefore extends the above results to the case of even space dimension. In addition, we present the parallel discussion for the group {0" 2 (t)} associated with the wave equation and zero Robin boundary condition in f2. We obtain the eigenfunction expansions for {Ul (t)} and {U2 (t)}, not as realizations of outgoing and incoming spectral representations, but through a separation of variables, modelled after the use of the Fourier transform in [1] in the construction of an eigenfunction expansion for the free-space wave equation. This construction, valid for arbitrary n, was based on the fact that the Fourier transform provides an eigenfunction expansion for the self-adjoint operator-A on L2 (R,), the plane waves {e-~ X'e: r(generalized eigenfunctions of the operator-A) giving the expansion.