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From (1.36) and D_n (∞) = 0, we have

(1.39)

and, since σ_n (x) is even in x, we have from (1.29),

(1.40)

It follows then from (1.34) and Figs. (1.38), (1.39) and (1.40),

(1.41)

which lead to, through (1.35),

(1.42)

and

(1.43)

Consider first the case that w (x) in (1.38) is positive and satisfies

(1.44)

The hierarchy theorem that will be proved in Section 3 states that if w (x) satisfies (1.44) then the iterative solution of (1.42) with the boundary condition

(1.45)

gives a convergent monotonic sequence , where for all n,

(1.46)

and

(1.47)

likewise, the sequence f₀ (x) = 1, f₁ (x), f₂ (x), … is also monotonic and convergent at any x   0 with

fn(x)>fn-1(x)

(1.48)

and

(1.49)

Furthermore, the convergence of Figs. (1.47) and (1.49) can hold for arbitrarily large but finite w (x). A result that is surprising, but pleasant.

On the other hand, if instead of (1.45), we impose a different boundary condition, one given by

(1.50)

then instead of (1.47), we have for all odd n = 2m + 1 an ascending sequence

(1.51)

however, for the even n = 2m series, we have a descending sequence

(1.52)

furthermore, between any even n = 2m and any odd n = 2l + 1, we have

(1.53)

Since according to (1.13), is the nth order iteration towards

(1.54)

each odd member in (1.51) gives an upper bound of E, whereas each even member in (1.52) leads to a lower bound of E. Both sequences approach the correct as n → ∞, one from above and the other from below. For the boundary condition (1.50), our proof of convergence requires a condition on the magnitude of w (x). Still this is quite a remarkable result.

In Section 2, we discuss the details of how to construct a good trial function  (q) for the N-dimensional problem. Section 3 gives the proof of the hierarchy theorem for the one-dimensional problem in which V (x) = V (−x) is an even function of x and the potential-difference function w (x) is assumed to satisfy (1.44); i.e., w′ (x) < 0 for x > 0. The extension to the asymmetric case V (x) ≠ V (−x) is discussed in Section 4. The hierarchy theorem is also applicable to Mathieu’s equation, which has infinite number of maxima and minima. In Appendix A, we give a soluble example in one dimension.

In dimensions greater than 1, at each iteration Eq. (1.21) gives a fine tuning of the energy, just like the one-dimensional problem. Hence, there are good reasons to expect our approach to yield convergent solutions in any higher dimension. In Section 5, we formulate an explicit conjecture to this effect. We describe an attempt to prove this conjecture by generalizing the steps used to prove the hierarchy theorem in one dimension. The attempt fails at present because the proof of one of the lemmas does not appear to generalize in higher dimension.

The present paper represents the synthesis and generalization of results, some of which have appeared in our earlier publications [1], [2], [3] and [4]. The function D_n introduced in this paper is identical to the function h_n used in [4].

2. Construction of trial functions

2.1. A new formulation of perturbative expansion

In many problems of interest, perturbative expansion leads to asymptotic series, which is not the aim of this paper. Nevertheless, the first few terms of such an expansion could provide important insight to what a good trial function might be. For our purpose, a particularly convenient way is to follow the method developed in [1] and [2]. As we shall see, in this new method to each order of the perturbation, the wave function is always expressible in terms of a single line-integral in the N-dimensional coordinate space, which can be readily used for the construction of the trial wave function.

We begin with the Hamiltonian H in its standard form (1.7). Assume V (q) to be positive definite, and choose its minimum to be at q = 0, with

V(q) V(0)=0.

(2.1)

Introduce a scale factor g² by writing

V(q)=g2v(q)

(2.2)

and correspondingly

ψ(q)=e-gS(q).

(2.3)

Thus, the Schroedinger equation (1.9) becomes

(2.4)

where, as before, q denotes q₁, q₂, … , q_N and the corresponding gradient operator. Hence S (q) satisfies

(2.5)

Considering the case of large g, we expand

S(q)=S0(q)+g-1S1(q)+g-2S2(q)+

(2.6)

and

E=gE0+E1+g-1E2+

(2.7)

Substituting Figs. (2.6) and (2.7) into (2.5) and equating the coefficients of g⁻ⁿ on both sides, we find

(2.8)

etc. In this way, the second-order partial differential equation (2.5) is reduced to a series of first-order partial differential equations (2.8). The first of this set of equations can be written as

(2.9)

As noted in [1], this is precisely the Hamilton–Jacobi equation of a single particle with unit mass moving in a potential “−v (q)” in the N-dimensional q-space. Since q = 0 is the maximum of the classical potential energy function −v (q), for any point q ≠ 0 there is always a classical trajectory with a total energy 0+, which begins from q = 0 and ends at the other point q ≠ 0, with S₀ (q) given by the corresponding classical action integral. Furthermore, S₀ (q) increases along the direction of the trajectory, which can be extended beyond the selected point q ≠ 0, towards ∞. At infinity, it is easy to see that S₀ (q) = ∞, and therefore the corresponding wave amplitude e^-gS₀^(q) is zero. To solve the second equation in (2.8), we note that, in accordance with Figs. (2.1) and (2.2) at q = 0, . By requiring S₁ (q) to be analytic at q = 0, we determine

(2.10)

It is convenient to consider the surface

S0(q)=constant;

(2.11)

its normal is along the corresponding classical trajectory passing through q. Characterize each classical trajectory by the S₀-value along the trajectory and a set of N − 1 angular variables

α=(α1(q),α2(q),…,αN-1(q)),

(2.12)

so that each α determines one classical trajectory with

αj· S0=0,

(2.13)

where

j=1,2,…,N-1.

(2.14)

(As an example, we note that as q → 0, and therefore . Consider the ellipsoidal surface S₀ = constant. For S₀ sufficiently small, each classical trajectory is normal to this ellipsoidal surface. A convenient choice of α could be simply any N − 1 orthogonal parametric coordinates on the surface.) Each α designates one classical trajectory, and vice versa. Every (S₀, α) is mapped into a unique set (q₁, q₂, … , q_N) with S₀   0 by construction. In what follows, we regard the points in the q-space as specified by the coordinates (S₀, α). Depending on the problem, the mapping (q₁, q₂, … , q_N) → (S₀, α) may or may not be one-to-one. We note that, for q near 0, different trajectories emanating from q = 0 have to go along different directions, and therefore must associate with different α. Later on, as S₀ increases each different trajectory retains its initially different α-designation; consequently, using (S₀, α) as the primary coordinates, different trajectories never cross each other. The trouble-some complications of trajectory-crossing in q-space is automatically resolved by using (S₀, α) as coordinates. Keeping α fixed, the set of first-order partial differential equation can be further reduced to a set of first-order ordinary differential equation, which are readily solvable, as we shall see. Write

S1(q)=S1(S0,α),

(2.15)

the second line of (2.8) becomes

(2.16)

and leads to, besides (2.10), also

(2.17)

where the integration is taken along the classical trajectory of constant α. Likewise, the third, fourth, and other lines of (2.8) lead to

(2.18)