A degenerate kernel method for eigenvalue problems of a class of non-compact operators

TL;DR

This paper introduces a degenerate kernel method for approximating eigenvalues of non-compact operators, providing exact matrix entries, convergence proof with rate O(h), and numerical validation.

Contribution

It presents a novel degenerate kernel approach for eigenvalue problems of non-compact operators, including exact matrix evaluation and convergence analysis.

Findings

Matrix entries can be evaluated exactly.

Convergence rate is proven to be O(h).

Numerical examples confirm theoretical results.

Abstract

We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method is used to approximate isolated eigenvalues. It is shown that entries of the corresponding matrix of this method can be evaluated exactly. The convergence of the method is proved; it is proved that the convergence rate is $O(h)$. By some numerical examples, we confirm the results.