Eigenvalues and eigenfunctions of a class of integral operators

Abstract

We present a method for determining analytically the eigenfunctions of a class of Fredholm integral operators, as well as the characteristic equation whose roots are equal to the eigenvalues of these operators. Using the generalized Leibniz rule, we are able to transform the eigenvalue equation for these integral operators into a second-degree, linear, nonhomogeneous differential equation, whose solutions are the eigenfunctions of this integral operator. Using the boundary conditions governing this integral operator, we then obtain the characteristic equation whose roots, computed numerically, are equal to the eigenvalues of the integral operator. Such a method can be applied in determining the eigenvalues and eigenfunctions of quantum time operators.