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Proof

Note that Figs. (5.34) and (5.35) are very similar to Figs. (3.56) and (3.57). As in (3.60), define

From (5.33), we have

and

Therefore,

where

Furthermore,

where

analogous to Figs. (3.61), (3.62), (3.63) and (3.64).

According to (5.30), at W = 0

and according to (5.29), at W = W_max

From (5.37), we see that the derivative is positive when , zero at , and negative when . Likewise, from (5.38), is positive when , zero at and negative when . Their ratio determines .

(i) If , from Lemma 1, we have

and therefore, on account of (5.42)

At W = 0,

As W increases, so does r (W). At , r (W) has a discontinuity, with

and

As W increases from , r (W) continues to increase, with

and

It is convenient to divide the range 0 < W < W_max into three regions:

Assuming , we shall show separately in these three regions.

In region B, Q_{l + 1} is decreasing, but Q_{m + 1} is increasing. Clearly,

In region A, , r (W) is positive according to Figs. (5.47) and (5.48) and is always >0 from (5.46). Therefore from (5.41),

In region C, , but r (W) and are both positive. Hence,

Within each region, η = Q_{m + 1} (W) and ξ = Q_{l + 1} (W) are both monotonic in W; therefore η is a single-valued function of ξ and we can apply Lemma 2 of Section 3.

In region A, at W = 0 both Q_{m + 1} (0) and Q_{l + 1} (0) are 0 according to (5.43), but their ratio is given by

Therefore

Furthermore, from (5.56), . It follows from Lemma 2 of Section 3, the ratio η/ξ is an increasing function of ξ. Since

we also have

In region C, at W = W_max, both Q_{m + 1} (W_max) and Q_{l + 1} (W_max) are 0 according to (5.44). Their ratio is

which gives at W = W_max

As W decreases from W_max to in region C, since , we have

Furthermore, from (5.57), in region C. It follows from Lemma 2 of Section 3, the ratio η/ξ is a decreasing function of ξ, which together with (5.64) lead to

Thus, we prove Case (i) of Lemma 3. Case (ii) of Lemma 3 follows from Case (i) by an exchange of the subscripts m and l. Lemma 3 is then proved.

So far, the above lk and lk are almost identical copies of lk and lk of Section 3, but now applicable to the N-dimensional problem. Difficulty arises when we try to generalize Lemma 4 of Section 3.

It is convenient to transform the Cartesian coordinates q₁, q₂, … , q_N to a new set of orthogonal coordinates:

with

and

where i or j = 1, 2, … , N − 1. Introducing

In terms of the new coordinates, the components of D_n are

Its divergence is

Combining (5.12) with (5.30), we have

therefore,

in which the integration is along the surface

From Figs. (5.11) and (5.71), it follows that

In terms of curvilinear coordinates, (5.7) can be written as

Substituting (5.76) into (5.74), we find

Because , (5.78) can also be written as

Here comes the difficulty. While the above Lemma 3 transfers relations between to those between Q_{m + 1}/Q_{l + 1}, the latter is

which is quite different from . This particular generalization of the lemmas in higher dimensions fails to establish the Hierarchy Theorem.

For the one-dimensional case discussed in Section 3, we have w′ < 0 and x 0; consequently (5.80) is . Therefore, Lemma 4 of Section 3 can also be established by using (5.80), and the proof of the Hierarchy Theorem can be completed.

References

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Appendix A. A soluble example

In this appendix, we consider a soluble model in which the potential V (x) of (4.67) is

with W² > μ² and

Following (4.68), we introduce two symmetric potentials:

with, for x 0,

and

so that (A.1) can also be written as

Let ψ (x), χ_a (x), and χ_b (x) be, respectively, the ground state wave functions of

and

For |x| > γ, since V (x) = ∞, we have

For |x| < γ, these wave functions are of the form

and

By substituting these solutions to the Schroedinger equations Figs. (A.7), (A.8) and (A.9), we derive

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