星形网络上色散波的多重隧道效应：谱表示的显式公式
Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation
摘要：
我们考虑由n个半轴在原点连接而成的星形网络上的Klein-Gordon方程。我们加上一个势，它是常数，但在每个分支上不同。相应的空间算子是自伴的，我们国家明确表示其预解式和其决议的身份广义本征函数。这导致广义傅立叶型反演公式的广义本征函数的展开。进一步我们证明了相关变换的满射性，从而表明它实际上是一个谱表示。该问题的特点是星形区域的非流形特征。因此，通过Sturm-Liouville系统理论的方法是不适合的。构建包含隧道效应广义本征函数的显式公式的相当大的努力是合理的，例如，通过研究隧道效应对L-无穷时间衰减的影响的角度。
英文摘要：
We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier type inversion formula in terms of an expansion in generalized eigenfunctions. Further we prove the surjectivity of the associated transformation, thus showing that it is in fact a spectral representation. The characteristics of the problem are marked by the non-manifold character of the star-shaped domain. Therefore the approach via the Sturm-Liouville theory for systems is not well-suited. The considerable effort to construct explicit formulas involving the tunnel effect generalized eigenfunctions is justified for example by the perspective to study the influence of tunnel effect on the L-infinity-time decay.
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