ABSTRACT
The strong limits from (5.17) of the Møller operators Ωˆ± are of central importance to scattering theory. One of the key illustrations in this direction can be provided by showing that the Lippmann-Schwinger equations for scattering states could be derived from these limits. To demonstrate this fact, we shall use the method of exponential screening of functions, as already explored in the adiabatic theorem for a special case of the operator potential interaction Vˆ. In general, the method of exponential attenuation of functions or operators can be formulated in the form of the following lemma: Lemma 4: Let ψ(t) ∈ H be a state vector for which the Laplace transform∫∞
0 dt e−εtψ(t) for ε > 0 exists and is given by
Lim t→∓∞ ψ(t) = ψ
±. (7.1)
Then the following equality is valid
∓Lim ε→0+ ε
dt e±ε tψ(t) = ψ±. (7.2)
Recalling (6.3), we see that the strong limit in (7.2) is the Abel limit, so that we can use the appropriate notation
∓Lim ε↓0 ε
dt e±ε tψ(t) = ψ± (ε > 0). (7.3)
The procedure of the Abel limiting process can be best understood by an example of an ordinary scalar function f(t) i.e. f(t) ∈ F , where F is a given scalar field. Then, we will assume that the limiting value of f(t) can be found when e.g. t→ −∞ and, as such, shall be denoted by f+. The statement (7.2) applies equally well to ordinary scalar functions as well as to abstract vectors of linear spaces. Therefore, we can say that the same value f+ should be attainable through the Abel limit
f+ = lim ε→0+
ε
dt eε t f(t). (7.4)
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Notice first that for the special case of the scalar function f(t) defined as a constant for every value of the independent variable t e.g. f(t) = f+, the elementary integration immediately yields the result
ε
dt eε t f+ = f+ ε ∫ 0 −∞
dt eε t = f+. (7.5)
However, if here the lower infinite bound (t = −∞) is replaced by a finite value T (−∞ < T < 0), then it easily follows that the finite integration region t ∈ [−T, 0] gives the zero contribution to the integral on the lhs of (7.5) when ε→ 0+, so that
ε
dt eε t f+ = f+ lim ε→0+
ε
dt eε t = lim ε→0+
= f+ lim ε→0+
( 1− e−ε|T |
) = 0
∴ lim ε→0+
ε
dt eε t f+ = 0. (7.6)
Next we consider a more general case where f(t) is not a constant function and has the limiting value f+ when t → −∞. Then in the limit ε → 0+ the whole contribution to the integral
∫ dt f(t) exp (εt) from the semi-infinite
interval t ∈ [−∞, 0] is still reduced to the term f+ alone, as can be shown by a simple change of variable
ε t = τ. (7.7)
This leads to
ε
dt eε t f(t) = lim ε→0+
dτ eτ f(τ/ε)
= ∫ 0 −∞
dτ eτ lim ε→0+
f(τ/ε)
= ∫ 0 −∞
dτ eτ f+ = f+ ∫ 0 −∞
dτ eτ = f+
∴ lim ε→0+
ε
dt eε t f(t) = f+ (QED) (7.8)
where we exchanged the order of the limiting process and integration. This is justified by the existence of underlying uniform convergence. From here we see that the concept of the strong limit of abstract vectors of elements of
the strong limits
a general linear space is an entirely natural extension of the usual uniform convergence from scalar functions. This is a very useful analogy. Let us now pass to a more general proof of lemma 4 for abstract elements
of vector spaces. For stationary vectors such as ψ(t) ≡ ψ− or ψ(t) ≡ ψ+ the equality (7.3) is reduced to the identity similarly to the corresponding case of a constant scalar function f(t) ≡ f+ or f(t) ≡ f−. In the case when ψ(t) is not a constant vector for any t, it is illustrative to carry out the analysis for e.g. ψ− = ∅. In such a case, strong convergence of ψ(t) means that for an arbitrary number ²′ > 0 we can choose T such that ||ψ(t)−ψ−|| = ||ψ(t)|| < ²′ whenever t > T. Thus, setting ²′ ≡ ²/2 (² > 0), we have for T > 0∥∥∥∥ε ∫ ∞
dt e−εtψ(t) ∥∥∥∥ = ε
∥∥∥∥∥ ∫ T 0
dt e−εtψ(t) + ∫ ∞ T
dt e−εtψ(t)
∥∥∥∥∥ ≤ ε
∥∥∥∥∥ ∫ T 0
dt e−εtψ(t)
∥∥∥∥∥ + ε ∥∥∥∥∫ ∞
dt e−εtψ(t) ∥∥∥∥
≤ ε ∫ T 0
dt e−εt ||ψ(t)|| + ε ∫ ∞ T
dt e−εt||ψ(t)||
≤ ε ∫ T 0
dt e−εt||ψ(t)|| + ε ² 2
∥∥∥∥∫ ∞ T
dt e−εt ∥∥∥∥
≤ ε ∫ T 0
dt e−εt||ψ(t)|| + ² 2 e−εT
≤ ε ∫ T 0
dt ||ψ(t)|| + ² 2 <
²
2 +
²
2 = ²
∴ ∥∥∥∥ε ∫ ∞
dt e−εtψ(t) ∥∥∥∥ < ² (QED). (7.9)
This proves strong convergence of the vector ψ(t) to ψ− = ∅, since the norm ||ε ∫∞
0 dt e−εt{ψ(t) − ψ−}|| is shown to be vanishingly small in the limit
ε −→ 0+. In the demonstration (7.9), we select ε < ²/(2 ∫ T 0
dt ||ψ(t)||) in the integral from T to ∞ and use the inequality ||ψ(t)|| ≤ ²′ ≡ ²/2 which is justified by the condition of lemma 4. We also use the Schwartz inequality for exp (−ε t) and exp (−ε T ) via | exp (−ετ)| = |1 − ετ + ε2τ2/2 − · · · | ≤ 1− ετ + ε2τ2/2− · · · ≤ 1 . Let us now apply lemma 4 to scattering states
Ψ(t) ≡ Ωˆ(t)Ψ0 (7.10)
Ωˆ(t) = Uˆ†(t)Uˆ0(t). (7.11)
Due to the presence of the evolution operators Uˆ and Uˆ0 and their underlying feature of isometry, it is possible to prove lemma 4 in a different way without
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considering two special cases with ψ+ being a non-zero vector and ψ+ = ∅. To this end, we shall first assume that the following strong limits exist
Ψ± = Lim t→∓∞ Ψ(t)
= Lim t→∓∞ Ωˆ(t)Ψ0
= Lim t→∓∞ Uˆ
= Lim t→∓∞ e
∴ Ψ± = Lim t→∓∞ Ψ(t) = Ωˆ
±Ψ0 (7.12)
where (5.3), (7.10) and (7.11) are used. Then lemma 4 asserts that there must also exist the Abel limit with the same limiting value as follows
Ψ± = ∓ Lim ε→0+ ε
d τ e±ετ Uˆ†(τ)Uˆ0(τ)Ψ0. (7.13)
Proof of lemma 4: Before proving that (7.13) follows from (7.12), we shall interpret the corresponding strong limits. For instance, the expression (7.12) means that ∥∥∥Uˆ†(t)Uˆ0(t)Ψ0 −Ψ−∥∥∥ < ²′ (7.14) for an arbitrary ²′ > 0 starting from t > T. A similar interpretation is valid for (7.13), since the Abel limit also represents strong convergence i.e. convergence in the norm. Namely, choosing an arbitrary number ² > 0, the norm of the difference between the state vectors from the lhs and rhs of (7.13) can be made as small as we desire i.e.∥∥∥∥ε∫ ∞
d τ e−ετ Uˆ†(τ)Uˆ0(τ)Ψ0 −Ψ− ∥∥∥∥ < ² (7.15)
when t becomes greater that certain T > 0. Splitting the τ−interval [0,∞] into two sub-intervals according to [0, T ] + [T,∞] , we shall have∥∥∥∥ε ∫ ∞
d τ e−ετ Uˆ†(τ)Uˆ0(τ)Ψ0 −Ψ− ∥∥∥∥ =
ε
∥∥∥∥∫ ∞ 0
d τ e−ετ{Uˆ†(τ)Uˆ0(τ)Ψ0 −Ψ−} ∥∥∥∥
≤ ε ∫ ∞ 0
d τ e−ετ‖|Uˆ†(τ)Uˆ0(τ)Ψ0 −Ψ−||
= ∫ ∞ 0
d τ e−τ ||Uˆ†(τ/ε)Uˆ0(τ/ε)Ψ0 −Ψ−||
the strong limits
=
{∫ T 0
+ ∫ ∞ T
} d τ e−τ ||Uˆ†(τ/ε)Uˆ0(τ/ε)Ψ0 −Ψ−||
≤ ∫ T 0
d τ e−τ{||Uˆ†(τ/ε)Uˆ0(τ/ε)Ψ0||+ ||Ψ−||}+ ²2 ∫ ∞ T
d τ e−τ
= ∫ T 0
d τ e−τ (||Ψ0||+ ||Ψ−||) + ²2 ∫ ∞ T
d τ e−τ
= (1− e−T )(||Ψ0||+ ||Ψ−||) + ²2e −T
≤ T (||Ψ0||+ ||Ψ−||) + ²2e −T
≤ T (||Ψ0||+ ||Ψ−||) + ²2 ≤ ²
2 + ²
2 = ²
∴ ∥∥∥∥ε ∫ ∞
d τ e−ετ Uˆ†(τ)Uˆ0(τ)Ψ0 −Ψ− ∥∥∥∥ ≤ ² (QED). (7.16)
Here, we used the expansion 1−exp (−T ) = 1− (1−T + · · · ) = T + · · · as well as the majorization exp (−ξ) ≤ 1 (ξ ≥ 0). Also in the integral over τ ∈ [T,∞] we employed the assumption (7.14) of lemma 4 for ε′ = ²/2. Additionally, we made the following choice of the otherwise arbitrary number T
T ≤ 1 2
²
||Ψ0||+ ||Ψ−|| . (7.17)
Applying lemma 4 to the state vector Ψ(t) from (7.10), we find
∓ε ∫ ∓∞ 0
dt e±εtΨ(t) = ∓ε ∫ ∓∞ 0
dt e±εt{eiHˆt e−iHˆ0tΨ0}
= ∓ε ∫ ∓∞ 0
dt e±εt{eiHˆt e−iEtΨ0}
= ∓ε ∫ ∓∞ 0
dt eit[Hˆ−(E∓iε)1ˆ]Ψ0
= ±iε[(E ± iε)1ˆ− Hˆ]−1Ψ0
∴ ∓ε ∫ ∓∞ 0
dt e±εtΨ(t) = ±iε[(E ± iε)1ˆ− Hˆ]−1Ψ0 (7.18)
where the integration is carried out by means of the spectral expansion of the unity operator in terms of the eigen-vectors of the Hamiltonian Hˆ, as in (6.10). Thus employing (7.13) and (7.18) we have
Ψ± = ±i Lim ε→0+ ε[(E ± iε)1ˆ− Hˆ]
−1Ψ0. (7.19)
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Further, the usage of the assumed separable form of the total Hamiltonian via Hˆ = Hˆ0 + Vˆ permits the arrival at the following transformation of the term ±iε[(E ± iε)1ˆ− Hˆ]−1Ψ0 from the rhs of (7.19)
±iε[(E ± iε)1ˆ− Hˆ]−1Ψ0 = ±iε[(E ± iε)1ˆ− Hˆ0 − Vˆ]−1Ψ0 = ±iε{[(E ± iε)1ˆ− Hˆ0][1ˆ− (E1ˆ− Hˆ0 ± iε1ˆ)−1Vˆ]}−1Ψ0 = ±iε{1ˆ− [(E ± iε)1ˆ− Hˆ0]−1Vˆ}−1[(E ± iε)1ˆ− Hˆ0]−1Ψ0 = ±iε{1ˆ− [(E ± iε)1ˆ− Hˆ0]−1Vˆ}−1(±iε)−1Ψ0 = {1ˆ− [(E ± iε)1ˆ− Hˆ0]−1Vˆ}−1Ψ0
∴ ±iε[(E ± iε)1ˆ− Hˆ]−1Ψ0 = {1ˆ− [(E ± iε)1ˆ− Hˆ0]−1Vˆ}−1Ψ0. (7.20)
Here, we made use of the following relation
± iε [(E ± iε)1ˆ− Hˆ0]−1Ψ0 = Ψ0 (7.21)
which is evident from the unperturbed eigen-problem (Hˆ0 −E1ˆ)Ψ0 = 0. Letting ε→ 0+ in (7.20) according to the requirement in (7.19), it follows
Ψ± = Lim ε→0+ {1ˆ− [(E ± iε)1ˆ− Hˆ0]
−1Vˆ}−1Ψ0 = [1ˆ− Gˆ±0 (E)Vˆ]−1Ψ0
∴ Ψ± = [1ˆ− Gˆ±0 (E)Vˆ]−1Ψ0. (7.22)
Since Ψ± = Ωˆ±Ψ0, as per (5.7), it is clear that by means of (7.22) we can readily identify the wave operators Ωˆ± in the forms
Ωˆ± = Lim ε→0+ {1ˆ− [(E ± iε)1ˆ− Hˆ0]
= [1ˆ− Gˆ±0 (E)Vˆ]−1. (7.23)
If both sides of (7.23) are multiplied by the operator 1ˆ − Gˆ±0 (E)Vˆ from the left, we will obtain the following two Lippmann-Schwinger equations for Ωˆ±
Ωˆ± = 1ˆ + Gˆ±0 (E)VˆΩˆ ±. (7.24)
When these Møller operators are applied to Ψ0 through (5.7), the LippmannSchwinger equations are obtained directly for scattering states Ψ± as
Ψ± = Ωˆ±Ψ0 = [1ˆ + Gˆ±0 (E)VˆΩˆ
±]Ψ0 = Ψ0 + Gˆ±0 (E)Vˆ{Ωˆ±Ψ0} = Ψ0 + Gˆ±0 (E)VˆΨ
the strong limits
∴ Ψ± = Ψ0 + Gˆ±0 (E)VˆΨ±. (7.25)
The same expression as in (7.25) can be obtained directly by applying the operator 1ˆ− Gˆ±0 (E)Vˆ from the right to both sides of (7.22). The Abel strong limit (7.13), which enabled the derivation of the Lippmann-Schwinger equation (7.25), is also capable of yielding the formal solution of these integral equations. This solution is contained in the intermediate step (7.19), since using the unperturbed eigen-problem (Hˆ0 − E1ˆ)Ψ0 = ∅, we have
Ψ± = ±iεGˆ±(E)Ψ0 = Gˆ±(E)iεΨ0 = Gˆ±(E)[(E ± iε)1ˆ− Hˆ0 − Vˆ + Vˆ]Ψ0 = Gˆ±(E)[(E ± iε)1ˆ− Hˆ + Vˆ]Ψ0 = [1ˆ + Gˆ±(E)Vˆ]Ψ0 = Ψ0 + Gˆ±(E)VˆΨ0
∴ Ψ± = Ψ0 + Gˆ±(E)VˆΨ0 (7.26)
in agreement with (6.12) and (6.13). This is the solution of (7.25), since the vector Ψ± is eliminated from the rhs of (7.26), unlike the situation in the corresponding integral equation (7.25). Nevertheless, the solution (7.26) is still of a purely formal nature since it is expressed through the total Green operator Gˆ±(E), which is just as difficult to handle as the original equation for the scattering states Ψ±. By combining the expressions (5.7) and (7.26), we can arrive at yet another useful representation of the Møller operators Ωˆ±
via the total Green resolvents Gˆ±(E)
Ωˆ± = 1ˆ + Lim ε→0+ [Hˆ− (E ∓ iε)1ˆ]
−1Vˆ
= 1ˆ + Gˆ±(E)Vˆ. (7.27)
If the identities Gˆ±(E) = Gˆ±0 (E)[1ˆ − VˆGˆ±0 (E)]−1 = [1ˆ − Gˆ±0 (E)Vˆ]−1Gˆ±0 (E) are multiplied by Vˆ, it will follow
Gˆ±0 (E)Tˆ ±(E) = Gˆ±(E)Vˆ (7.28)
Tˆ±(E)Gˆ±0 (E) = VˆGˆ ±(E) (7.29)
where Tˆ±(E) are the transition Tˆ−operators Tˆ±(E) = [1ˆ− VˆGˆ±0 (E)]−1Vˆ = Vˆ[1ˆ− Gˆ±0 (E)Vˆ]−1. (7.30)
Using (7.28), it is possible to express Ωˆ± via the transition Tˆ−operators according to
Ωˆ± = 1ˆ + Gˆ±0 (E)Tˆ ±(E). (7.31)
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The accomplished analysis completes the proof that the strong limit (5.9) leads to the time-independent i.e. stationary Lippmann-Schwinger integral equations that represent the cornerstone of the quantum scattering theory. From the performed analysis it is clear that all the results are independent
of the manner of averaging in the integral representations of the Møller wave operators. The same expression (7.25), as well as the existence of the wave operator Ωˆ+ can be proven by employing the following averaging
Ωˆ+ = Lim T→+∞
1 T
dt Uˆ†(t)Uˆ0(t). (7.32)
With this formula, the results for Ωˆ± appear as a generalization of the wellknown Neumann mean ergodic theorem. The results from (7.25) can also be obtained from the same starting point
as we just did, but proceeding afterwards in a different way. Namely, let us take the intermediate result (7.19), which can be rewritten via the total Green resolvent Gˆ(z) from (6.15) as
Ψ± = ±i Lim ε→0+ ε Gˆ(E ± iε)Ψ0. (7.33)
If the obvious identity
(z − z′)1ˆ = (z1ˆ− Hˆ)− (z′1ˆ− Hˆ) = Gˆ−1(z)− Gˆ−1(z′ ) (7.34) is multiplied from the left by Gˆ(z′ ), it follows
Gˆ(z′ )− Gˆ(z) = (z − z′ )Gˆ(z)Gˆ(z′ ) = (z − z′ )Gˆ(z′ )Gˆ(z). (7.35)
On the other hand, the relation Hˆ = Hˆ0 + Vˆ can be expressed via
Vˆ = Hˆ− Hˆ0 = (z1ˆ− Hˆ0)− (z1ˆ− Hˆ) = Gˆ−10 (z)− Gˆ−1(z) (7.36) where Gˆ0(z) is the free Green resolvent (6.15). Multiplications of (7.36) from the left by Gˆ0(z) and from the right by Gˆ(z) or in the opposite order, lead to the Lippmann-Schwinger equation for the resolvent operator Gˆ(z)
Gˆ(z) = Gˆ0(z) + Gˆ0(z)VˆGˆ(z) = Gˆ0(z) + Gˆ(z)VˆGˆ0(z). (7.37)
With the help of (7.37), the relation (7.33) now becomes
Ψ± = ± i Lim ε→0+ ε Gˆ0(E ± iε)Ψ0
± i Lim ε→0+ ε Gˆ0(E ± iε) Vˆ Gˆ(E ± iε)Ψ0. (7.38)
Further, in terms of Gˆ0(z), the expression (7.21) acquires the form
±iε Gˆ0(E ± iε)Ψ0 = Ψ0 (7.39)
the strong limits
so that by means of (7.38), we have
Ψ± = ± i Lim ε→0+ ε Gˆ0(E ± iε)Ψ0
± i Lim ε→0+ Gˆ0(E ± iε)Vˆε Limε→0+ Gˆ(E ± iε)Ψ0
= Ψ0 + Lim ε→0+ Gˆ0(E ± iε)Ψ
≡ Ψ0 + Gˆ0(E ± i0)VˆΨ± = Ψ0 + Gˆ±0 (E)VˆΨ±
∴ Ψ± = Ψ0 + Gˆ±0 (E)VˆΨ± (7.40) where (7.33) is used. The obtained expression (7.40) is in full agreement with the previously derived Lippmann-Schwinger equations for scattering states Ψ± from (7.25). Notice that the signs of the terms ±ε in (7.39) coincide with the order of the
superscripts ± in the Møller wave operators Ωˆ± and the scattering states Ψ± from the same equations. Such an ordering of the signs ± in Ωˆ± is in accord with the superscripts of the Green operators Gˆ±(E) = Lim
ε→0+ Gˆ(E ± iε) in (6.15) where the signs ± have a physical interpretation relative to the corresponding boundary conditions for the outgoing/incoming waves, respectively. Now such an interpretation is via (7.38) directly transferred into the Green Gˆ±(z) and Møller Ωˆ± operators. Their superscripts ± are introduced in (5.4) so as to correspond to the limits t → ∓∞ of the operator Ωˆ(t) from (7.11), and this also explains an apparently illogical convention (5.4).
