%0 Journal Article %1 vkaplitskii11may:eigenf %A Kaplitskii, V M %D 2011 %J IZV MATH %K eigenfunction estimate operator unread %N 5 %P 933-958 %R 10.1070/IM2011v075n05ABEH002559 %T A method for estimating eigenfunctions of integral operators of certain classes in unbounded domains %V 75 %X We describe a method for obtaining estimates at infinity for eigenfunctions of integral operators of certain classes in unbounded domains of $ 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)ınØmega\timesØmega$, where $ k_0(x,y)\vertłeþeta(x-y)e^-S(x-y)$ for some functions $ þeta$ 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(Ømega)$ and the coefficients $ a(x)$, $ b(y)$ satisfy certain conditions, then the solutions of $ \varphi=K\varphi$ belong to the weighted space $ L_p(Ømega, e^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.

@article{vkaplitskii11may:eigenf, abstract = {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 $ \vert k_0(x,y)\vert\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.}, added-at = {2011-11-13T13:06:55.000+0100}, author = {Kaplitskii, V M}, biburl = {https://www.bibsonomy.org/bibtex/28026dc020cd6a09537bec7a30ab761b2/drmatusek}, doi = {10.1070/IM2011v075n05ABEH002559}, interhash = {746afc630f083540960b4838972055c9}, intrahash = {8026dc020cd6a09537bec7a30ab761b2}, journal = {IZV MATH}, keywords = {eigenfunction estimate operator unread}, month = {sep-oct}, number = 5, pages = {933-958}, timestamp = {2012-05-21T13:51:49.000+0200}, title = {A method for estimating eigenfunctions of integral operators of certain classes in unbounded domains }, volume = 75, year = 2011 }