Internal eigenstate problem: the trial state method

It is shown that the device of adding a special trial state to a basis set, thus augmenting by 1 the dimension of any complex matrix being studied, leads to a formalism that permits the application of the wave operator approach to calculating the internal spectrum of the matrix as well as the action of the resolvent operator (E-H)(-1) on an arbitrary vector in the original N-dimensional space. Two calculational variants of the method are described and both are tested by studying the problem of a short laser pulse interacting with a H+2 ion.