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Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 5, Pages 65–92 (Mi izv4100)  

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

V. M. Kaplitskii

Southern Federal University

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 $|k_0(x,y)|\le\theta(x-y)e^{-S(x-y)}$ 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.

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

DOI: https://doi.org/10.4213/im4100

Full text: PDF file (638 kB)
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English version:
Izvestiya: Mathematics, 2011, 75:5, 933–958

Bibliographic databases:

UDC: 517.984
MSC: 47G10, 45C05, 47A53, 35P20, 35Q40
Received: 24.03.2009
Revised: 03.03.2010

Citation: V. M. Kaplitskii, “A method for estimating eigenfunctions of integral operators of certain classes in unbounded domains”, Izv. RAN. Ser. Mat., 75:5 (2011), 65–92; Izv. Math., 75:5 (2011), 933–958

Citation in format AMSBIB
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