Approximation of eigen values and eigenvectors

Many realistic problems that arise in theoretical physics lead to equations which cannot be given exact general solutions. Therefore, many attempts have been made to produce satisfactory methods to provide approximate solutions. The most common ones search for solutions in terms of power series and in terms of asymptotic expansions. The main idea behind approximation methods is to try to find approximate solutions of a given system by means of exact solutions of an approximate system (a ‘nearby’ system). We start with the eigenvalue equation associated with the Hamiltonian H and consider an approximating one, say H0, whose equations of motion can be explicitly solved. We introduce an interpolating family of Hamiltonians Hε = H0 + εH1 such that, for a given value of ε, say ε, Hε ≡ H. Both eigenvalues and eigenvectors for Hε may be expanded as a series in ε and solved order by order. The simplest example to illustrate what we have in mind is provided by a second-order algebraic equation, e.g.

x2 − 3.03x + 2.02 = 0.

This equation may be written by means of an interpolating family

x2 − 3(1 + ε)x + 2(1 + ε) = 0,

which yields the equation we want to solve when ε = 10−2. One can consider the approximate equation to be given by ε = 0, i.e.

x2 − 3x + 2 = 0.