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As in (4.2), ψ (x) is the ground state wave function that satisfies

which can also be written in the same form as (1.14)

with

Here, unlike (1.32), V (x) can now also be asymmetric. Taking the difference between ψ (x) times (4.75) and χ (x) times (4.80), we derive

Introduce

in which f (x) satisfies

On account of Figs. (4.82) and (4.83), the same equation can also be written as

Eq. (4.80) will again be solved iteratively by introducing

with ψ_n and its associated energy determined by

and

In terms of f_n (x), we have

On account of (4.88), we also have

and

For definiteness, let us assume that

in Figs. (4.69a) and (4.69b); therefore and , in accordance with (4.76a). Start with, for n = 0,

we can derive {E_n} and {f_n (x)}, with

by using the boundary conditions, either

or

It is straightforward to generalize the Hierarchy theorem to the present case. As in Section 3, in Case (A), the validity of the Hierarchy theorem imposes no condition on the magnitude of . But in Case (B) we assume to be not too large so that (4.91) and the boundary condition f_n (−∞) = 1 is consistent with

for all finite x. From the Hierarchy theorem, we find in Case (A)

and

while in Case (B)

and

A soluble model of an asymmetric square-well potential is given in Appendix A to illustrate these properties.

5. The N-dimensional problem

The N-dimensional case will be discussed in this section. We begin with the electrostatic analog introduced in Section 1. Suppose that the (n − 1)th iterative solution f_{n − 1} (q) is already known. The nth order charge density σ_n (q) is

in accordance with Figs. (1.23) and (1.24). Likewise, from Figs. (1.26) and (1.29) the dielectric constant κ of the medium is related to the trial function (q) by

and the nth order energy shift is determined by

In the following we assume the range of w (q) to be finite, with

and

Introduce

where δ (w (q) − W) is Dirac’s δ-function, W is a constant parameter and the integrations in Figs. (5.3) and (5.6) are over all q-space. Similarly, for any function F (q), we define

In the N-dimensional case, the generalization of [F], introduced by (3.15), is

In terms of F^av (W), (5.8) can also be written as

Thus from Figs. (5.1) and (5.3) we have

the n-dimensional extension of (3.14).

Following Figs. (1.27) and (1.28), the nth order electric field is and the displacement field is

The corresponding Maxwell equation is

Eqs. Figs. (5.11) and (5.12) determine f_n except for an additive constant, which can be chosen by requiring

Therefore,

As in the one-dimensional case discussed in Section 3, (5.10) gives the same condition of fine energy tuning at each order of iteration. It is this condition that leads to convergent iterative solutions derived in Section 3. We now conjecture that

and

also hold in higher dimensions. Although we are not able to establish this conjecture, in the following we present the proofs of the N-dimensional generalizations of some of the lemmas proved in Section 3.

Lemma 1

For any pair f_m(q) and f_l(q) if at all W within the range (5.5),

and

Proof

For any function , define

Thus for any function F (q), we have

therefore,

and

By setting the subscript n in (5.10) to be m + 1, we obtain

Also by definition (5.19),

The difference of Figs. (5.23) and (5.24) gives

From (5.10) and setting the subscript n to be l + 1, we have

Regard and as two constant parameters. Multiply (5.25) by , (5.26) by and take their difference. The result is

analogous to (4.43).

(i) If , then for

Thus, the function inside the bracket in (5.21) is positive, being the product of two negative factors, and . Also, when , these two factors both reverse their signs. Consequently (5.17) holds.

(ii) If , we see that for , (5.28) reverses its sign, and therefore the function inside the bracket in (5.27) is now negative. The same negative sign can be readily established for . Consequently, (5.18) holds and Lemma 1 is established.

Lemma 2

Identical to Lemma 2 of Section 3.

In order to establish the N-dimensional generalization of Lemma 3 of Section 3, we define

Because of (5.3), Q_n (W) is also given by

We may picture that the entire q-space is divided into two regions

and

with Q_n (W) the total charge in I, which is also the negative of the total charge in II. By using Figs. (5.1) and (5.7), we see that

Lemma 3

For any pair f_m (q) and f_l (q) if at all W within the range (5.5)

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