Analytic Properties of Moments Matrices

Associated with each matrix element of the recently developed Generalized Moments Expansion, \textrm{GMX}$(n,m)$ there is a unique expansion for the ground state energy in terms of the \textquotedblleft connected moments\textquotedblright\ $I_{k}$ of the Hamiltonian (Phys. Lett.~\textbf{A}% 349, 320 [2006]). That is, for any set $\{n,m\}$ a polynomial in the $I_{k}$'s may be generated to any desired order $L$, which is dependent upon the highest moment calculated. Here we wish to study the eigenvectors and eigenvalues of the \textrm{GMX} matrix itself. Furthermore we investigate the interplay between the set $\{n,m\}$ and the order $L$ of the matrix in determining which combination $\{n,m,L\}$ yields the \textquotedblleft best\textquotedblright% \ (i.e.~most convergent) result for the ground state energy.