Analyticity of parametric elliptic eigenvalue problems and applications to quasi-Monte Carlo methods

In the present paper, we study analyticity of leftmost eigenvalue linear elliptic partial differential operators with random coefficient and analyse convergence rate quasi-Monte Carlo method for approximation expectation this quantity. The is assumed to be represented by an affine expansion a0(x)+∑j∈Nyjaj(x), where elements parameter vector y=(yj)j∈N∈U∞ are independent identically uniformly distributed on U:=[−12,12]. Under assumption ‖∑j∈Nρj|aj|‖L∞(D)<∞ some positive sequence (ρj)j∈N∈ℓp(N) p∈(0,1] show that any y∈U∞, operator has a countably infinite number eigenvalues (λj(y))j∈N which can ordered non-decreasingly. Moreover, spectral gap λ2(y)−λ1(y) in U∞. From this, prove holomorphic extension property λ1(y) complex domain C∞ estimate derivatives respect y using Cauchy's formula analytic functions. Based these bounds dimension-independent approximate λ1(y).