Is Perturbation Theory Enough for Relativistic Corrections?

In some recent articles in the literature (1),(2) the Schrodinger equation:

                    p*p/2m W(x) + V(x) W(x) = E W(x) ((1))

is replaced with an equation using a relativistic momentum operator:

[sqrt(p*p + m*m) - m] W(x) + V(x)W(x) = E W(x) ((2))

The first operator on the LHS of ((2)) is then expanded in powers of 1/(m*m). Next, p is replaced with hbar/i d/dx (or gradient in 3D) and perturbation theory used.

In this note, we consider p*p as being a conditional average given by:

  p*p= [Integral dp  f(p) exp(ipx) p*p ] /  [Integral dp f(p) exp(ipx)]    ((3))

The denominator is the wavefunction W(x).

This result should hold for the nonrelativistic case. In the relativistic case, p is replaced everywhere with prel where prel= p/ sqrt(1-p*p/(m*m))

prel*prel = Integral dprel   p*p (1+ p*p/(m*m)) f(prel) exp(iprel x)   ((4))

And W(x) = Integral dprel f(prel) exp(iprel x)    ((5))

To first order, we argue that Integral dprel f(prel) exp(iprel x) = Integral dp f(p) exp(i p x), but Integral p*p f(prel) dprel exp(i prel x) contributes a term . For higher orders, even Integral dprel f(prel) exp(i prel x) contributes.  Thus, we try to argue that one does not only have a change in the prel*prel operator, but one must consider changes of prel within Integral dprel f(prel) exp(iprel x) itself. Thus, perturbation theory may not be enough in such cases.