Criteria of exponential decay for eigenfunctions of some classes of integral operators

Abstract

We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ n . We consider integral operators K whose kernels have the form $$ k\left( {x,y} \right) = c\left( {x,y} \right)\frac{{{e^{ - \alpha \left| {x - y} \right|}}}}{{{{\left| {x - y} \right|}^\beta }}},\,\left( {x,y} \right) \in \Omega \times \Omega, $$ where α > 0, 0 ≤ β < n, and c(x, y) ∈ L ∞(Ω × Ω). It is shown that if the corresponding operator T = I − K is Fredholm, then all the solutions of the integral equation φ = Kφ exponentially decay at infinity. We consider applications to Wiener–Hopf operators with oscillating coefficient and to some classes of convolution operators with variable coefficients. Bibliography: 14 titles.