A method for estimating eigenfunctions of integral operators of certain classes in unbounded domains

V. M. Kaplitskii

Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 933-958

Résumé

We describe a method for obtaining estimates at infinity for eigenfunctions of integral operators of certain classes in unbounded domains of $\mathbb{R}^n$. We consider integral operators $K$ whose kernels $k(x,y)$ can be written in the form $k(x,y)=a(x)k_0(x,y)b(y)$, $(x,y)\in\Omega\times\Omega$, where $|k_0(x,y)|\le\theta(x-y)e^{-S(x-y)}$ for some functions $\theta$ and $S$ satisfying certain natural additional conditions. We show that if the operator $T=I-K$ with the corresponding kernel is Noetherian in $L_p(\Omega)$ and the coefficients $a(x)$, $b(y)$ satisfy certain conditions, then the solutions of $\varphi=K\varphi$ belong to the weighted space $L_p(\Omega, e^{\delta S(x)})$. The method is applied to obtain exponential estimates for eigenfunctions of $N$-particle Schrödinger operators and estimates of decay at infinity for the solutions of convolution-type equations with variable coefficients.

Détail

Keywords: integral operator, Noetherian operator, eigenfunction, exponential decay, discrete spectrum.