We study the statistics of Dirichlet eigenvalues of the random Schr "odinger operator @math , with @math the discrete Laplacian on @math and @math uniformly bounded independent random variables, on sets of the form @math for @math bounded, open and with a smooth boundary. If @math holds for some bounded and continuous @math , we show that, as @math , the @math -th eigenvalue converges to the @math -th Dirichlet eigenvalue of the homogenized operator @math , where @math is the continuum Dirichlet Laplacian on @math . Assuming further that @math for some positive and continuous @math , we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation. The limiting covariance for a given pair of simple eigenvalues is expressed as an integral of @math against the product of squares of the corresponding eigenfunctions of @math .
As alluded to earlier, a result closely related to ours has been derived by Bal @cite_17 . There the operator of the form @math in @math with Dirichlet boundary condition is studied, where @math is a random centered stationary field. Note that this can naturally be regarded as a spatially scaled version of our model. (Bal in fact studied the more general situation where @math is replaced by a pseudo differential operator.) In dimensions @math and under the assumptions that either @math is bounded and has an integrable correlation function, or @math and a mixing condition holds ([H2] on page 683 of @cite_17 ), it is proved in Section 5.2 that the @math -th smallest eigenvalue @math of @math has Gaussian fluctuations around @math with @math , provided this eigenvalue is simple. This is slightly different from our result, which shows a CLT around the expectation. In the case @math , we know that @math by combining the result of Bal with ours, but we do not know how to prove this directly.
@cite_17
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We consider the perturbation of elliptic pseudodifferential operators @math with more than square integrable Green's functions by random, rapidly varying, sufficiently mixing, potentials of the form @math . We analyze the source and spectral problems associated with such operators and show that the rescaled difference between the perturbed and unperturbed solutions may be written asymptotically as @math as explicit Gaussian processes. Such results may be seen as central limit corrections to homogenization (law of large numbers). Similar results are derived for more general elliptic equations with random coefficients in one dimension of space. The results are based on the availability of a rapidly converging integral formulation for the perturbed solutions and on the use of classical central limit results for random processes with appropriate mixing conditions.