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Relation between the Three‐Dimensional Fredholm Determinant and the Jost Functions
1.R. Jost and A. Pais, Phys. Rev. 82, 840 (1951).
2.See Ref. 1, and for further discussion, for example, Ref. 3.
3.R. G. Newton, Scattering Theory of Waves and Particles (McGraw‐Hill, New York, 1966).
4.It is easy to see that for this equation is equivalent to the statement that is an eigenvalue of because is positive definite.
5.G. Hoheisel, Integral Equations (Ungar, New York, 1968), p. 79;
5.also W. Pogorzelski, Integral Equations (Pergamon, Oxford, 1966), Vol. I, p. 178.
6.See Ref. 3, p. 345 and p. 375 for the proof that the zero is simple as a function of E. Equation (9.10) on p. 236 of Ref. 3 shows that it is simple as a function of γ.
7.The same remark as Ref. 4 applies here.
8. is a spherical harmonic as a function of the polar angles θ and φ.
9.It should be noted that whereas never exists, no matter how well behaved the potential, on exists if the first absolute moment of V is finite.
10.See, for example, Sec. 9. 1 of Ref. 3.
11.See Refs. 1 and 3.
12.See Chap. 12 of Ref. 3.
13.Because is an analytic function of γ.
14.That the pole of the resolvent is simple at a simple zero of the Fredholm determinant follows immediately from the Fredholm construction. That there is no degeneracy if the determinant has a simple zero follows from the theorem of the Appendix.
15.The ascent of an operator A is the smallest integer s such that the nullspaces of and are equal. Since is compact, the ascent of is finite; see, for example, A. E. Taylor, Introduction to Functional Analysis (Wiley, New York, 1958), p. 279. For a demonstration that the ascent of equals the order of the pole of the resolvent at see, for example, Ref. 3, p. 208.
16.See the corollary stated in the Appendix.
17.The resolvent may be decomposed according to the decomposition of into a direct sum of the eigenspaces of the angular part of the Laplacean. Thus , where are projections whose kernels are .
18.This theorem is well known for finite‐dimensional matrices; for Hilbert‐Schmidt operators it does not seem to appear in the literature.
19.“Tr” denotes the trace of the operator. A projection (not necessarily orthogonal) of finite‐dimensional range is in the trace class and its trace equals the dimensionality of its range.
20.The proof of (A12) from (A11) follows from the fact that if K is in the trace class, then Equation (A12) therefore holds for K in the trace class, with both sides defined as absolutely convergent power series in γ. Since neither side contains TrK and (A12) is an identity of the two power series whose convergence is assured if K is a Hilbert‐Schmidt operator, it must hold also if TrK does not exist.
21.The intersection of the range of with the nullspace of L equals the image under of the complement of the nullspace of relative to the nullspace of Since the range of is a non‐increasing function of m, the nullspace of must be strictly increasing until the nullspaces of and of are equal. From then on they must remain equal as m increases.
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