On the Connection Between the Resolvent Methods and Partitioning Technique, Rational Approximations, and Perturbation Theory

In the resolvent methods for solving the Schrödinger equation, the eigenvalues are represented by simple poles of the so-called Weinstein function, whereas the eigenfunctions are represented by expressions of the form ∞/∞. It is shown that there exists a simple identity which permits the evaluation of the eigenfunction, and which shows the connection between the resolvent methods and partitioning technique, rational approximations, and perturbation theory. It is also shown that this identity may be extended to the solution of the Liouville equation.