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A key quantity that occurs in the error analysis of several numerical methodsfor eigenvalue problems is the distance between the eigenvalue of interest andthe next nearest eigenvalue. When we are interested in the smallest orfundamental eigenvalue, we call this the spectral or fundamental gap. In arecent manuscript [Gilbert et al., arXiv:1808.02639], the current authors,together with Frances Kuo, studied an elliptic eigenvalue problem withhomogeneous Dirichlet boundary conditions, and with coefficients that depend onan infinite number of uniformly distributed stochastic parameters. In thissetting, the eigenvalues, and in turn the eigenvalue gap, also depend on thestochastic parameters. Hence, for a robust error analysis one needs to be ableto bound the gap over all possible realisations of the parameters, and becausethe gap depends on infinitely-many random parameters, this is not trivial. Thisshort note presents, in a simplified setting, an important result that wasshown in the paper above. Namely, that, under certain decay assumptions on thecoefficient, the spectral gap of such a random elliptic eigenvalue problem canbe bounded away from 0, uniformly over the entire infinite-dimensionalparameter space.

Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients