The eigenvalue spectrum in quantum mechanics and the nonlinearization procedure

A new approach to the eigenvalue problem in quantum mechanics is proposed. This approach is based on three propositions: 1) a perturbation theory which does not require knowledge of the entire eigenvalue spectrum of the unperturbed problem and which uses a “nonlinearization” procedure (leaving some latitude in the choice of a zeroth order approximation); 2) a relationship between the perturbation theory and a variational principle, namely that any variational calculation is none other than the first two terms of some nontrivial perturbation theory which, when developed further, can reveal the accuracy of the variational calculations and can refine them by an iterative procedure; 3) “Dyson's argument”, which serves as a criterion for the “reasonableness” of the choice of a zeroth order approximation (the unperturbed problem). The realization of this perturbation theory in a $k$-dimensional space is equivalent to the solution of a $k$-dimensional electrostatic problem with a variable dielectric permittivity. In the one-dimensional case and in cases which reduce to the one-dimensional case, all the corrections are written in quadratures. It is shown that the construction of an ordinary perturbation theory (in which the zeroth order approximation is an exactly solvable problem) within the framework of this perturbation theory is a purely algebraic procedure, which reduces to the solution of some simple recurrence relations. An approximation analogous to the leading logarithmic approximation of quantum field theory is constructed. Some standard problems of quantum mechanics–the anharmonic oscillator, the Zeeman effect, and the Stark effect–are treated as examples. It is shown that this new approach makes it possible to develop systematically a theory for strong coupling and large perturbations.