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

V. M. Kaplitsky

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 37, Tome 366 (2009), pp. 53-66

Résumé

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 $\alpha>0$, $0\leq\beta$, $c(x,y)\in L_\infty(\Omega\times\Omega)$. 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.