Eigenvalue and Eigenfunction Statistics
In the foregoing section, we considered in detail statistical characteristics of the wavefield in random medium. We discussed the problems on the wave incident on a medium layer (half-space) and the waves generated by a source in the medium. In parallel with the above problems, physics of disordered systems (see, e.g., ) places high emphasis on studying the statistics of eigenvalues of the Helmholtz equation (energy levels of the Schrödinger equation) for bounded randomly inhomogeneous systems. Wave propagation in different waveguides is an additional example of such problems (see, e.g., ). In the general case of many-dimensional systems, the analysis of eigenvalue and eigenfunction statistics faces great difficulties. However, in the one-dimensional case (plane layered media) the consideration appears significantly simpler.
KeywordsIndicator Function Riccati Equation Helmholtz Equation Planck Equation Liouville Equation
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