Bound States—Eigenfunction Expansions
The bound states correspond to the solutions of (1.1.6) which satisfy the boundary conditions (1.2.1), and are square-integrable on the whole positive r-axis. We are going to study them in detail because, as is well known, they are necessary in the completeness relation, and therefore, as was mentioned before, enter into the inverse problem. We again consider the S-wave first. The assumptions on the potential are those of the last section of chapter I : rV(r) ∈ L1(b, ∞), b > 0, W(r) ∈ L1(0, a), and rW(r) → 0 as r → 0, which are, evidently, weaker than the usual assumption rV(r) ∈ L1(0, ∞).
KeywordsPhase Shift Integral Representation Asymptotic Form High Wave Eigenfunction Expansion
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