A method for obtaining bounds on eigenvalues and eigenfunctions by solving non-homogeneous integral equations

A new technique is presented for obtaining upper and lower bounds on eigenvalues and eigenfunctions for linear integral equations. The method is unique in that the bounds are obtained by solving non-homogeneous equations. In order to solve the non-homogeneous equations, non-linear sequence-to-sequence transformations are used to accelerate convergence of the Neumann series inside the radius of convergence and are used to “sum” the Neumann series outside the radius of convergence. Since the reciprocals of the eigenvalues appear as poles in the solution of the non-homogeneous equation, a very sensitive bounding criterion can be given. The method applies to quite general kernels, and has been successfully applied to symmetric and non-symmetric kernels. In addition, thek ^th eigenfunction may be obtained without a knowledge of the first (k?1) eigenvalues or eigenfunctions.