Excited states in logarithmic perturbation theory (Q1091530)

The authors discuss cases for which formal solutions of the one dimensional Schrödinger equation \[ \psi ''(x)+[\lambda -U_ 0(x)- \epsilon U_ 1(x)]\psi (x)=0,\quad x\in (a,b), \] can be obtained from the substitution \(\phi (x)=\exp [-y(x)]\) and expansions \[ y'(x)=v_ 0+\epsilon v_ 1+\epsilon^ 2v_ 2+...,\quad \lambda =\lambda_ 0+\epsilon \lambda_ 1+\epsilon^ 2\lambda_ 2+.... \] A difficulty of this method arises if the unperturbed state \(\psi_ 0(x)=\exp [-y_ 0(x)]\) has zeros within (a,b). At such points, it turns out that the \(v_ n\) can go to \(+\infty\) and complicated procedures must be applied. The authors notice that there are cases where these complicated methods can be by passed. This is the case if the potentials \(U_ 0\) and \(U_ 1\) have appropriate symmetry, if the perturbation \(U_ 1\) takes place in a subdomain of (a,b) or if \(\psi_ 0\) is nonzero. All these cases are discussed; in particular a general criterion for \(\psi_ 0\) to be nonzero is given.