# Bound States—Eigenfunction Expansions

Chapter

## Abstract

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) ∈ L*^{1}(b, ∞), *b >* 0, W(r) ∈ L^{1}(0, *a)*, and *rW(r) → 0 as r → 0, which are, evidently, weaker than the usual assumption rV(r) ∈ L*^{1}(0, ∞).

## Keywords

Phase Shift Integral Representation Asymptotic Form High Wave Eigenfunction Expansion
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## Copyright information

© Springer-Verlag New York Inc. 1989