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

## TL;DR

This paper establishes that, under certain conditions, the spectral gap of a stochastic elliptic eigenvalue problem can be uniformly bounded away from zero across all infinite-dimensional parameter realizations, aiding robust error analysis.

## Contribution

It provides a simplified proof that the spectral gap remains uniformly positive for a class of stochastic elliptic eigenvalue problems with decaying coefficients.

## Key findings

- Spectral gap can be bounded away from zero uniformly over all stochastic parameters.
- The result applies under decay assumptions on the coefficient.
- Supports robust error analysis for stochastic eigenvalue problems.

## Abstract

A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental eigenvalue, we call this the spectral or fundamental gap. In a recent manuscript [Gilbert et al., arXiv:1808.02639], the current authors, together with Frances Kuo, studied an elliptic eigenvalue problem with homogeneous Dirichlet boundary conditions, and with coefficients that depend on an infinite number of uniformly distributed stochastic parameters. In this setting, the eigenvalues, and in turn the eigenvalue gap, also depend on the stochastic parameters. Hence, for a robust error analysis one needs to be able to bound the gap over all possible realisations of the parameters, and because the gap depends on infinitely-many random parameters, this is not trivial. This short note presents, in a simplified setting, an important result that was shown in the paper above. Namely, that, under certain decay assumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue problem can be bounded away from 0, uniformly over the entire infinite-dimensional parameter space.