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

We investigate conditions sufficient for an exponential decay of eigenfunctions in the case of a certain class of integral equations in unbounded domains in $\mathbb R^n$. The integral operators $K$ in question have kernels of the form
$$ k(x,y)=\frac{c(x,y)}{|x-y|^\beta}\,e^{-\alpha|x-y|},\qquad x,y\in\Omega\subset\mathbb R^n, $$
where $\alpha>0$, $0\leq\beta<n$, $c(x,y)\in L_\infty(\Omega\times\Omega)$. It is shown that, if the operator $T=I-K$ is Fredholm, then all solutions of the equation $\varphi=K\varphi$ have exponential decay at infinity. Applications to Wiener–Hopf operators with oscillating coefficient and some classes of convolution operators with variable coefficients are considered. Bibl. – 14 titles.