7.7 We demonstrated that the S-matrix is unitary and that its poles correspond to \(L^{2}\)-normalizable eigenstates of the Hamiltonian, at least in the case of the narrow step potential. Assuming this is true in general for an arbitrary localized and
\footnotetext{
8 H. von Koch, "Sur une courbe continue sans tangente, obtenue par une construction géométrique
} élémentaire," Ark. Mat. Astron. Fys. 1, 681-702 (1904).

compact potential, there are a few things that we immediately know about the structure of the S-matrix.
(a) Can an S-matrix have poles at real values of momentum \(p\) ? If yes, provide an example; if no, explain why not.
(b) What is the large-momentum limit of the \(\mathrm{S}\)-matrix? That is, as \(|p| \rightarrow \infty\), what is the "worst" scaling with \(p\) that the S-matrix can exhibit?

Transcribed Image Text:

Vo V(x) I+ -a 0 Fig. 7.19 Plot of a potential with a finite barrier of height Vo and width a and a hard, infinite potential for x > 0.