A NEW APPROACH TO INVERSE SPECTRAL THEORY, III. SHORT-RANGE POTENTIALS

ALEXANDER RAMM^{1} AND BARRY SIMON^{2}

1. Introduction

In this paper we discuss aspects of the spectral theory of half-line Schrödinger operators

Hu = −u′′ + qu,  (1.1)

where q ∈ L_{loc}^{1}([0, ∞)). Indeed, all the q’s considered in this paper will lie in L^{1}([0, ∞), dx). The Weyl m-function m(z) is defined for Im z > 0 by looking at the solution u(x; z) of −u′′ + qu = zu which is L^{2} at infinity (unique up to constants if q is limit point and unique once a boundary condition at infinity is chosen if q is limit circle) and setting

m(z) = \frac{u′(0; z)}{u(0; z)} .  (1.2)

In [10], we considered the fundamental object A(α) associated to q by the relation

m(−κ^{2}) = −κ − ∫_{0}^{∞} A(α) e^{−2ακ} dα,  (1.3)

where A(α) ∈ L^{1}(0, a) for all a. Formula (1.3) holds in the sense of an absolutely convergent integral if Re κ is sufficiently large in case q ∈ L^{1} ([10]) or if q ∈ L^{∞} ([2]). For general q, it holds in the same sense that a series is asymptotic to a function ([2],[10]).

1Mathematics Department, Kansas State University, Manhattan, KS 66506, USA. E-mail: ramm@math.ksu.edu

2Division of Physics, Mathematics, and Astronomy, 253-37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: bsimon@caltech.edu. This material is based upon work supported by the National Science Foundation under Grant No. DMS-9707661. The Government has certain rights in this material.

Date: May 11, 1999.

1991 Mathematics Subject Classification. Primary: 34B25, 34A55.

Key words and phrases. Weyl function, spectral theory, inverse problems.

To appear in J. d’Analyse Math.

In [10], it is proven that if q ∈ L^{1}(0, ∞), then

|A(α) − q(α)| ≤ ∥q∥_{1}^{2} exp(α∥q∥_{1}).

As explained in [10], A is fundamental to an approach to inverse spectral theory. Our goal in this paper is to study A as an interesting object in its own right and, in particular, using ideas implicit in Ramm [6] to obtain detailed information on the behavior of A as α → ∞ when q decays sufficiently fast as x → ∞. Indeed, for potentials decaying rapidly enough, Ramm [6] stated the representation (1.3), but no proof was given (nor was there any connection of the function A to the inverse problem for q. In [6] the inverse problem of finding the potential from the knowledge of the m-function has been solved for short-range potentials. A more detailed discussion of the result in [6] can be found in [7]).

Throughout this paper, we will suppose that

∫_{0}^{∞}(1 + |x|)|q(x)| dx < ∞.  (1.4)

More generally, we will consider for n = 0, 1, 2, …, B ≤ 0 and ℓ ≥ 0, the space C_{n}^{B,ℓ} of all functions f with n − 1 classical derivatives and f^{(n)} ∈ L^{1}(0, ∞) so that

∫_{0}^{∞}(1 + |x|)^{ℓ} e^{−Bx}|q^{(j)}(x)| dx < ∞  (1.5)

for j = 0, 1, …, n. Thus (1.4) says q ∈ C_{n=0}^{B=0,ℓ=1}.

Under condition (1.4), general principles (see, e.g., [5]) imply that for all κ with Re κ ≥ 0, there is a unique solution f(x, κ) of −f′′ + qf = −κ^{2}f so that f(x, κ) = e^{−κx}(1 + o(1)) as x → ∞. We will set f(κ) ≡ f(x = 0, κ). Except for the change of variables κ = −ik, f(x, κ) and f(κ) are the standard Jost solution and Jost function. Both f(x, κ) and f(κ) are analytic in κ in {κ | Re κ > 0}. If q ∈ C_{n}^{B,ℓ} for any n, ℓ and B < 0, then f(x, κ) and f(κ) have analytic continuations into the region Re κ > B/2 (see Section 2 below).

It is easy to see and well known that [5]

(1) The zeros of f in {κ | Re κ > 0} occur precisely at those points κ_{j} with −κ_{j}^{2} a bound state of the operator H with u(0) = 0 boundary condition and each zero is simple.

(2) f has no zeros on {κ | Re κ = 0, κ ≠ 0}.

(3) If f(0) = 0 and q ∈ C_{n=0}^{B=0,ℓ=2}, then f is C^{1} and f′(0) ≠ 0. If f(0) = 0, we say that q has a zero energy resonance.

If f can be analytically continued to {κ | Re κ > B/2} for B < 0, then zeros of f in {κ | Re κ < 0} are called resonances. They occur in complex conjugate pairs (since f is real on the real axis). If f′(κ_{0}) ≠ 0 at a zero κ_{0}, we say that κ_{0} is a simple resonance. Resonances need not be simple if Re κ_{0} < 0 although they are generically.

The result stated in [6] can be phrased:

Theorem 1.1. Suppose that q obeys (1.4) (lies in C_{n=0}^{B=0,ℓ=1}) and does not have a zero energy resonance. Let {−κ_{j}^{2}}_{j=1}^{J} be the negative eigenvalues of H (with u(0) = 0 boundary condition) with κ_{j} > 0. Then

A(α) = ∑_{j=1}^{J} B_{j} e^{2ακ_{j}} + g(α),  (1.6)

where g ∈ L^{1}(0, ∞). In particular, if H has no bound states or zero energy resonance (e.g., if q ≥ 0), then A ∈ L^{1}.

Remarks. 1. The result stated in [6] assumes implicitly that there is no zero energy resonance. Details can be found in [7].

2. If A ∈ L^{1}, then (1.3) can be analytically continued to the entire region Re κ ≥ 0.

3. If u_{j}(x) is the eigenfunction for H, at energy −κ_{j}^{2}, normalized so that ∫_{0}^{∞} |u_{j}(x)|^{2} dx = 1, then

B_{j} = −\frac{|u′_{j}(0)|^{2}}{κ_{j}} .  (1.7)

This follows from the relation between A and the spectral measure [2]

A(α) = −2 ∫_{−∞}^{∞} λ^{−1/2} sin(2α\sqrt{λ}) dρ(λ)

and the fact that dρ(λ) ↾ (−∞, 0) = ∑_{j=1}^{J} |u′_{j}(0)|^{2}δ(λ + κ_{j}^{2}) dλ.

To handle zero energy resonances, one needs an extra two powers of decay (just as (1.4) says more or less that |q(x)| is bounded by O(x^{−2−ε}), the condition in the next theorem says that |q(x)| is more or less O(x^{−4−ε})):

Theorem 1.2. Let q ∈ C_{n=0}^{B=0,ℓ=3}. Suppose that H (with u(0) = 0 boundary condition) has a zero energy resonance and negative eigenvalues at {−κ_{j}^{2}}_{j=1}^{J} with κ_{j} > 0. Then

A(α) = B_{0} + ∑_{j=1}^{J} B_{j} e^{2ακ_{j}} + g(α),  (1.8)

where g ∈ L^{1}(0, ∞).

These results are special cases of Theorem 1.3. Let q ∈ C_{n}^{B=0,ℓ} where ℓ ≥ 1 and, if q has a zero energy resonance, then ℓ ≥ 3. Then (1.6) (resp. (1.8) if there is a zero energy resonance) holds where g ∈ C_{n}^{B=0,ℓ−1} (resp. C_{n}^{B=0,ℓ−3}).

Remark. If n ≥ 1, then g′ ∈ L^{1} so g → 0 pointwise. In general, if q is not continuous, then g may not go to zero. For example, let q(x) = ∑_{j=0}^{∞} χ_{[n,n+a_{n}]}(x) where a_{n} = e^{−n^{2}} and χ_{[a,b]} is the characteristic function of [a, b]. Since A(α) − q(α) is continuous, we are guaranteed that A(α) has jump discontinuities at α = n and so \overline{lim} A(α) − \underline{lim} A(α) ≥ 1 and thus, lim A(α) cannot be zero (since it cannot exist). In this case, A = g since there are no bound states or zero energy resonance.

For B < 0, we will prove:

Theorem 1.4. Let q ∈ C_{n}^{B,ℓ=0} with B < 0. Let B̃ ∈ (B, 0) and let {−κ_{j}^{2}}_{j=1}^{J} with κ_{j} > 0 be the negative eigenvalues, {λ_{j}}_{j=1}^{M} with λ_{j} ≤ 0 the real resonances (a.k.a. anti-bound states) and {μ_{j} ± iν_{j}}_{j=1}^{N} the complex resonances with B̃ ≤ μ_{j} < 0 with ν_{j} > 0. Suppose each resonance is simple. Then for suitable {B_{j}}_{j=1}^{J}, {C_{j}}_{j=1}^{M}, {D_{j}}_{j=1}^{N}, {θ_{j}}_{j=1}^{N}, we have that

A(α) = ∑_{j=1}^{J} B_{j} e^{2ακ_{j}} + ∑_{j=1}^{M} C_{j} e^{2αλ_{j}} + ∑_{j=1}^{N} D_{j} e^{2μ_{j}α} cos(2ν_{j}α + θ_{j}) + g̃(α),

where g̃(α) ∈ C_{n}^{B̃,ℓ=0}. In particular, if H has no negative eigenvalues, the rate of decay of A(α) is determined by the resonance with the least negative value of λ or μ.

In Section 4, we will discuss what happens when there are non-simple resonances.

We note that in the appendix, we present a result on principal ideals in the space of Laplace transforms (Corollary A.5) that may be of interest in its own right.

We thank F. Gesztesy for useful comments.

2. The Levin-Marchenko Representation

The key to the proof of Theorems 1.1–1.4 is the formula

m(−κ^{2}) = \frac{\frac{∂f(x,κ)}{∂x} |_{x=0}}{f(x = 0, κ)}  (2.1)

for the solution f of

−f′′ + qf = −κ^{2}f  (2.2)

with

f(x, κ) = e^{−κx}(1 + o(1))  (2.3)

as x → +∞.

We use a basic Laplace transform representation for f(x, κ) found by Levin [3] and developed especially by Marchenko [4],[5] and the theory of Laplace transforms discussed in the appendix.

The Levin-Marchenko representation says that given any q ∈ C_{n=0}^{B=0,ℓ=1} = {q | ∫_{0}^{∞} x|q(x)| dx < ∞}, there is a continuous function K(x, y) defined on {(x, y) | 0 ≤ x ≤ y < ∞} with ∫_{x}^{∞} |K(x, y)| dy < ∞ for each x so that

f(x, κ) = e^{−κx} + ∫_{x}^{∞} K(x, y) e^{−κy} dy.  (2.4)

Define

σ(x) = ∫_{x}^{∞} |q(y)| dy  (2.5)

so ∫_{0}^{∞} x|q(x)| dx < ∞ is equivalent to σ ∈ L^{1}. Then the following bound on K is known ([5], Lemma 3.1.1):

|K(x, y)| ≤ Cσ (\frac{x + y}{2} )  (2.6)

for some constant C. Indeed, one can take C = \frac{1}{2} exp(∫_{0}^{∞} σ(y) dy). Moreover, K is absolutely continuous in each variable. Indeed, if H(u, v) is defined in {(u, v) | 0 ≤ v ≤ u} by

H(u, v) = K(u − v, u + v)  (2.7a)
K(x, y) = H(\frac{1}{2}(x + y), \frac{1}{2}(y − x)),  (2.7b)

then H obeys (see [5], proof of Lemma 3.1.2):

\frac{∂H}{∂u} (u, v) = −\frac{1}{2} q(u) − ∫_{0}^{v} q(u − β)H(u, β) dβ  (2.8)
\frac{∂H}{∂v} (u, v) = ∫_{u}^{∞} q(α − v)H(α, v) dα.  (2.9)

From these bounds and equations, we have by a straightforward estimate that:

Theorem 2.1. If q ∈ C_{n}^{B,ℓ} with ℓ ≥ 1 and B ≤ 0, then

(i) K(x = 0, ·) ∈ C_{n+1}^{B,ℓ−1}
(ii) \frac{∂K}{∂x} (x = 0, ·) ∈ C_{n}^{B,ℓ−1}.

Remark. In fact, if B < 0, we can replace ℓ − 1 by ℓ.

3. Proof of the Main Theorems

Theorems 1.3 and 1.4 will follow from Theorem 2.1, formulas (2.1), (2.4), and the results in the appendix. Let

F (κ) ≡ f(x = 0, κ) = 1 + ∫_{0}^{∞} K(x = 0, y) e^{−κy} dy.  (3.1)

By Theorem 2.1, F ∈ 𝔄_{n+1}^{B,ℓ−1} with F (∞) = 1. The Banach algebra 𝔄_{n}^{B,ℓ} is discussed in the appendix. Thus, by Theorem A.8

F (κ)^{−1} = ∑_{ℓ=1}^{J̃} \frac{a_{ℓ}}{κ − κ̃_{ℓ}} + G(κ),

where

(i) If B < 0, {κ_{ℓ}}J̃ ; ℓ=1 are the zeros of F on Re κ > B and G ∈ 𝔄_{n+1}^{B̃,ℓ−1} for any B̃ > B.

(ii) If B = 0 and F (0) ≠ 0, then G ∈ 𝔄_{n+1}^{B=0,ℓ−1}. (iii) If B = 0 and F (0) = 0, we need ℓ ≥ 3 and then G ∈ 𝔄_{n+1}^{B=0,ℓ−3}.

Remarks. 1. For (i), we suppose all resonances are simple.

2. For (ii), (iii), by general principles, the zeros in Re κ ≥ 0 occur only for Im κ = 0. Moreover, each such zero is simple (see Lemma 3.1.6 of [5]).

Moreover, since K is C^{1} in y, by an integration by parts, F (κ) = 1 + K(0, 0)/κ + O(1/κ^{2}), and thus

F (κ)^{−1} = 1 − \frac{K(0, 0)}{κ} + O (\frac{1}{κ^{2}} ) .  (3.2)

Indeed, by Theorem A.7,

κF (κ)^{−1} = κ − K(0, 0) + ∑_{ℓ=1}^{J̃} \frac{a_{ℓ}κ_{ℓ}}{κ − κ_{ℓ}} + G̃(κ),  (3.3)

where G̃ is the same 𝔄 space as G, except that the number of derivatives is n, not n + 1. Moreover, G̃(∞) = 0.

Let

H(κ) ≡ \frac{∂}{∂x} f(x, κ)|_{x=0} = −κ − K(0, 0) − ∫_{0}^{∞} \frac{∂K}{∂x} (0, y)|_{x=0} e^{−κy} dy
≡ −κ + H̃(κ),

where H̃ lies in 𝔄_{n}^{B,ℓ−1} and H̃(∞) = −K(0, 0).

Therefore, by (3.2),

m(−κ^{2}) = −κF (κ)^{−1} + H̃(κ)F (κ)^{−1}
= −κ + ∑_{ℓ=0}^{J̃} b_{ℓ}(κ − κ_{ℓ})^{−1} + M̃(κ),  (3.4)

where M̃(∞) = 0 (since the K(0, 0) terms cancel) and M̃ ∈ 𝔄_{n}^{B̃,ℓ_{0}} with ℓ_{0} = ℓ − 3 (if B = 0 and F (0) = 0), and otherwise ℓ_{0} = ℓ − 1 and B̃ = 0 (if B = 0), and otherwise B̃ ∈ (B, 0). Since (κ − κ_{ℓ})^{−1} = 2 ∫_{0}^{∞} e^{2κ_{ℓ}t}e^{−2κt} dt if Re κ > Re κ_{ℓ}, Theorems 1.3 and 1.4 are proven.

4. Extensions and Remarks

It is only for simplicity that we supposed the resonances were simple. By iterating Theorem A.4 if f ∈ 𝔄_{n}^{B,ℓ} and Re z_{0} > B and z_{0} is a zero of F of order j, then g(z) = F (z)/(z − z_{0})^{j} is in 𝔄_{n+j}^{B,ℓ}. Thus, if some zeros of f(κ) are of order j, (3.4) need only be modified so that there are terms

∑_{ℓ=0}^{J̃} ∑_{j=1}^{ν̄} b_{ℓ}^{j}(κ − κ_{ℓ})^{−j},

where ν̄ = max(order of zeros in the strip). Since

\frac{1}{(κ − κ_{ℓ})^{j}} = \frac{2}{(j − 1)!} ∫_{0}^{∞} α^{j−1} e^{−2ακ}e^{2ακ_{ℓ}} dα,

we obtain an explicit formula for A like in Theorem 1.4, but C_{j} e^{2αλ_{j}} is replaced by a polynomial in α times e^{2αλ_{j}} and D_{j} e^{2μ_{j}α} cos(2ν_{j}α + θ_{j})

is replaced by ∑ν̃ ; k=0 d_{j,k}α^{k} e^{ν,α} cos(2ν_{j}α + θ_{j}^{(k)}).

If q ∈ 𝔄_{n=0}^{B} for all B, for example, q is compactly supported, it is known that there are always infinitely many resonances (see [8], pp. 280–282, [9], [11]). Thus A(α) does not decay faster than exponentially in these cases. It is natural to conjecture that A(α) never decays faster than exponentially as α → ∞ for q ≢ 0.

Appendix A. On the Theory of Laplace Transforms

In this appendix, we present some basic facts about Laplace transforms of functions that we need in the paper itself. Given the vast literature on Banach algebras, it may be that these results are presented elsewhere, but since we know of no precise reference, we present what is needed here.

Definition. A weight is a strictly positive function, w(α), on [0, ∞) that obeys:

(a) w(0) = 1.

(b) log w(α) is concave.

(c) For some α_{0} and real A_{0}, w(α) ≥ e^{−A_{0}α} for α ≥ α_{0}.

Lemma A.1. Let w be a weight. Then: (i)

w(α + β) ≤ w(α)w(β)  (A.1)

(ii) 1/α log w(α) is monotone decreasing (iii) A(w) ≡ − inf_{α} 1/α log w(α) = − lim_{α→∞} 1/α log w(α) < ∞ (iv) w(α) e^{A(w)α} is monotone increasing; in particular,

w(α) ≥ e^{−A(w)α}  (A.2)

for all α.

Proof. (i) By (a) and (b), w(α) ≥ w(α + β)^{α/α+β} and w(β) ≥ w(α + β)^{β/α+β}. (A.1) results by taking the product of these relations.

(ii) If α < β, then by (a) and (b),

w(α) ≥ w(β)^{α/β}.

(iii) is a consequence of (c).

(iv) Given α < β < γ, we have by (b) that

w(β) ≥ w(α)^{(γ−β)/(γ−α)}w(γ)^{(β−α)/(γ−α)}.

By (iii), lim_{γ→∞} w(γ)^{1/γ} = e^{−A(w)} so as γ → ∞, we have that

w(β) ≥ w(α) e^{−A(β−α)},  (A.3)

which is the required monotonicity. The final statement is just w(α) e^{A(w)α} ≥ w(0) e^{A(w)0}.

Remarks. 1. For many facts below, only (A.1) is critical, but we will need (A.3) in one place, so requiring log concavity seemed reasonable.

2. The prime example of weights is

w_{B,ℓ}(α) = (1 + α)^{ℓ}e^{−Bα}  (A.4)

for any B ∈ ℝ and ℓ ≥ 0. Notice that A(w_{A,ℓ}) = B. Definition. Let n be a non-negative integer and w a weight. We define C_{n,w} to be those complex valued functions f ∈ L^{1}([0, ∞), w(α) dα) with n distributional derivatives (viewed as distributions on C_{0}^{∞}(0, ∞)) f′, …, f^{(n)} all in L^{1}([0, ∞), w(α) dα) (so C_{n=0,w} is L^{1}(w dα)). If f ∈ C_{n,w} with n ≥ 1, then f is C^{n−1} in classical sense and lim_{α↓0} f^{(ℓ)}(α) exists for ℓ = 0, 1, …, n − 1 which we denote by f^{(ℓ)}(0). We norm C_{n,w} with

∥f∥_{n,w} = ∑_{ℓ=0}^{n−1} |f^{(ℓ)}(0)| + ∫_{0}^{∞} [∑_{ℓ=0}^{n} |f^{(ℓ)}(α)|] w(α) dα.  (A.5)

We denote C_{n,w_{B,ℓ}} by C_{n}^{B,ℓ} where w_{B,ℓ} is given by (A.4). As usual, the convolution of two L_{loc}^{1} functions on [0, ∞) is given by

(f ∗ g)(α) = ∫_{0}^{α} f(β)g(α − β) dβ.

By induction, it is easy to see that if f, g ∈ C_{n,w}, then for 1 ≤ ℓ ≤ n,

(f ∗ g)^{(ℓ)}(α) = ∑_{j=0}^{ℓ−1} f^{(j)}(0)g^{(ℓ−1−j)}(α) + (f^{(ℓ)} ∗ g)(α).

This formula and (A.1) imply that C_{n,w} ∗ C_{n,w} ⊂ C_{n,w} and, by (A.5)

∥f ∗ g∥_{n,w} ≤ ∥f∥_{n,w}∥g∥_{n,w}  (A.6)

so that C_{n,w} is a Banach algebra (without unit) under convolution.

Given μ ∈ ℂ and f ∈ C_{n,w}, we define the Laplace transform L(μ, f)(z) for z ∈ D(A(w)) ≡ ∞ ∪ {z | Re z ≥ A(w)} by

L(μ, f)(z) = μ + ∫_{0}^{∞} f(α) e^{−αz} dα (z ≠ ∞)  (A.7)
= μ  (z = ∞).

By (A.2), the integral in (A.7) converges absolutely and defines a function analytic in D(A(w))^{int} and continuous on D(A(w)). We denote the set of functions F (z) = L(μ, f)(z) for some f ∈ C_{n,w} by 𝔄_{n,w}. For w = w_{B,ℓ}, we denote 𝔄_{n,w} by 𝔄_{n}^{B,ℓ}. We norm 𝔄_{n,w} via |||L(μ, f)|||_{n,w} = |μ| + ∥f∥_{n,w}. Since

L(μ, f)L(λ, g) = L(μλ, μg + λf + f ∗ g),

(A.6) shows that 𝔄_{n,w} is a Banach algebra with unit.

We will need the following result of Wiener type:

Theorem A.2. Suppose F ∈ 𝔄_{n,w} obeys F (z) ≠ 0 for all z in D(A(w)). Then G(z) ≡ 1/F (z) is also in 𝔄_{n,w}.

Results of Wiener type are usually proven via the Gelfand theory of commutative Banach algebras (see [1], Ch. III for a good exposition). Using those ideas, Theorem A.2 is a direct consequence of

Proposition A.3. Every multiplicative linear functional on 𝔄_{n,w} is of the form F ↦ F (z) for z fixed in D(A(w)).

Proof. Let φ : 𝔄_{n,w} → ℂ be a multiplicative linear functional. Since L(1, 0) is the identity, one has φ(L(1, 0)) = 1, so φ is determined by the functional T (f) = φ(L(0, f)) on the space C_{n,w}. We claim that T is determined by its values on C_{0}^{∞}(0, ∞) ⊂ C_{n,w}. If n = 0, this is obvious since C_{0}^{∞}(0, ∞) is then dense in C_{n,w}. But if n > 0, \overline{C_{0}^{∞}(0, ∞)} is not all of C_{n,w}, so the argument is more subtle.

Indeed, if n ≥ 1, any f in C_{n,w} is C^{n−1} in the classical sense and f ∈ \overline{C_{0}^{∞}(0, ∞)} has f(0) = f′(0) = ⋯ = f^{(n−1)}(0) = 0. In fact, it is not hard to see that \overline{C_{0}^{∞}(0, ∞)} has codimension n − 1 in C_{n,w}.

Now for any g ∈ C_{n,w}, g^{n+1} (n+1-fold convolution) vanishes at 0 with its first n−1 derivative zero, and so it lies in \overline{C_{0}^{∞}(0, ∞)}. If T (g^{n+1}) = 0, then T (g) = 0, and if T (g^{n+1}) ≠ 0, then T (g) = T (g^{n+2})/T (g^{n+1}). Either way, T (g) is determined by T on C_{0}^{∞}(0, ∞).

The map g ↦ T (g) for g ∈ C_{0}^{∞}(0, ∞) defines a distribution which we denote by T (α). Formally, T (f) = ∫_{0}^{∞} T (α)f(α) dα and

T (α + β)(f(α)g(β)) = ∫_{0}^{∞} ∫_{0}^{∞} dα dβ T (α + β)f(α)g(β).

The idea of the proof is to derive the functional equation (A.8) for T (α) and to show that the distribution T (α) is in fact a regular distribution corresponding to a smooth function. If this is done, then one derives from (A.8) that T (α) is an exponential. If this is done, the proof is easy to complete.

The convolution formula for the Laplace transform implies T (f ∗g) = T (f)T (g). Since T (α + β)(f(α)g(β)) = T (f)T (g) and, since the linear span of the set of products of functions f(α)g(β) is dense in the set of functions C_{0}^{∞}(R_{+} × R_{+}) where R_{+} := (0, ∞), one concludes that

T (α + β) = T (α)T (β)  (A.8)

in distributional sense. If T ≢ 0, then T (f) ≠ 0 for some f ∈ C_{0}^{∞}. Thus,

T (β) = ∫_{0}^{∞} T (α + β) f(α) dα/T (f)  (A.9)

so T is a C^{∞} function. Thus, (A.8) implies that T′(α)/T (α) is constant and then that T (α) = e^{−zα} for some z ∈ ℂ (or else T ≡ 0).

Suppose that Re z < A(w). Then w(α) ≤ C_{1} exp(−\frac{1}{2}(A(w)+Re z)α) and thus, f(α) ≡ α^{n}e^{zα} lies in C_{n,w}. But if η ∈ C_{0}^{∞}(0, ∞), then T (ηf) = ∫_{0}^{∞} T (α)η(α)e^{zα}α^{n} dα = ∫_{0}^{∞} α^{n}η(α) dα. Taking η a smoothed out characteristic function of (δ, δ^{−1}), and taking δ → 0, has η_{δ}f → f in C_{n,w} but T (η_{δ}f) → ∞. Thus, Re z ≥ A(w), that is, z ∈ D(A(w)). Thus every φ is an evaluation at some z ∈ D(A(w)). Every such evaluation clearly defines a multiplicative linear functional.

We also need to know about factoring out zeros.

Theorem A.4. Let z_{0} ∈ D(A(w))^{int} and let F ∈ 𝔄_{n,w} obey F (z_{0}) = 0. Then G(z) = F (z)/(z − z_{0}) lies in 𝔄_{n+1,w}.

Proof. Suppose first that F (∞) = 0 so F = L(0, f) for some f in C_{n,w}. Let Ã = Re z_{0} +1 > A(w) so f ∈ CÃ ; n. Let H(z) = (z −z_{0})^{−1} = L(0, h), where

h(α) = e^{αz_{0}}

so h ∈ C_{n}^{Ã,ℓ=0}. Thus in D(Ã), G(z) = L(0, g) with g = h ∗ f, that is,

g(α) = ∫_{0}^{α} h(α − β)f(β) dβ
= ∫_{0}^{α} e^{(α−β)z_{0}}f(β) dρ  (A.10)
= − ∫_{α}^{∞} e^{(α−β)z_{0}}f(β) dβ

since F (z_{0}) = 0 means ∫_{0}^{∞} e^{−βz_{0}}f(β) dβ = 0.

Thus,

∫_{0}^{∞} |g(α)|w(α) dα ≤ ∫_{0}^{α} (∫_{α}^{∞} e^{(α−β) Re z_{0}}|f(β)| dβ) w(α) dα
= ∫_{0}^{∞} (∫_{0}^{β} e^{(α−β) Re z_{0}}w(α) dα) |f(β)| dβ. (A.11)

By Lemma A.1(iv) for α ≤ β:

e^{(α−β) Re z_{0}}w(α) ≤ e^{(α−β) Re z_{0}}e^{−A(w)(α−β)}w(β)

= e^{−(α−β)(A(w)−Re z_{0})}w(β).

Since A(w) Re z_{0} < 0,

∫_{0}^{β} e^{−(α−β)(A−Re z_{0})} dα = ∫_{0}^{β} e^{(A−Re z_{0})γ} dγ  (A.12)

is bounded by −(A − Re z_{0})^{−1} independent of β so (A.11) shows that

∫_{0}^{β} |g(α)|w(α) dα ≤ (Re z_{0} − A)^{−1} ∫_{0}^{α} |f(β)|w(β) dβ  (A.13)

and thus g ∈ L^{1}(w(α) dα).

By (A.6) for 1 ≤ ℓ ≤ n + 1,

g^{(ℓ)}(α) = ∑_{j=0}^{ℓ−1} z_{0}^{j}f^{(ℓ−1−j)}(α) + z_{0}^{ℓ} g(α).  (A.14)

(A.13) and (A.14) therefore show that g ∈ C_{n+1,w} as required.

For general F with F (∞) = μ, pick z_{1} with Re z_{1} < A(w) and define F_{0} by

F (z) = μ \frac{z − z_{0}}{z − z_{1}} + F_{0}(z).  (A.15)

Since

\frac{z − z_{0}}{z − z_{1}} = 1 + (z_{0} − z) ∫_{0}^{∞} e^{αz_{1}}e^{−αz} dα

and e^{· z_{1}} ∈ C_{n,w} for all n, we see F_{0}(z) ∈ 𝔄_{n,w} with F_{0}(∞) = F_{0}(z_{0}) = 0. Thus, G(z) = μ(z − z_{1})^{−1} + G_{0}(z) lies in 𝔄_{n,w}.

This theorem has an interesting corollary (immediate consequence of Theorem A.2):

Corollary A.5. Let F ∈ 𝔄_{n,w} be non-vanishing on D(A(w)) except for zeros at z_{1}, …, z_{m} ∈ D(A(w))^{int} of order ℓ_{1}, …, ℓ_{m}. Then for any z_{0} ∉ D(A(w)),

F (z) = G(z) ∏_{j=1}^{m} (\frac{z − z_{j}}{z − z_{0}} )^{ℓ_{j}} ,

where G is non-vanishing on all of D(A(w)) and lies in 𝔄_{k,w}. In particular, G is invertible, and any two such F ’s generate the same principal ideal.

We will need to know what happens where z_{0} ∈ ∂D(A(w)). First, we will consider z_{0} ≠ ∞ when we lose some decay of the inverse Laplace transform.

Theorem A.6. Let w be a weight and define the weight

w̃(α) = (1 + |α|)w(α).  (A.16)

Let Re z_{0} = A(w) and let F ∈ 𝔄_{n,w̃} obey F (z_{0}) = 0. Then G(z) = F (z)/(z − z_{0}) lies in 𝔄_{n+1,w}.

Remark. Notice that F ∈ 𝔄_{n,w̃} but G ∈ 𝔄_{n−1,w} with w, not w̃, so we lose one degree of decay for L^{−1}(F). For w_{B,ℓ}, this is not surprising. Since F is analytic in D(A(w))^{int}, dividing by (z − z_{0}) does not lose smoothness of F but since F is only C^{ℓ} on ∂D(A(w)), we expect the division to lose one order of smoothness.

Proof. As in the proof of Theorem A.4, (A.15) lets us consider the case F (∞) = 0 and (A.14) lets us reduce to the case n = 0. In that case, one follows the proof of Theorem A.4, but since −A(w) = Re z_{0}, (A.12) becomes

∫_{0}^{β} dα = β,

so (A.13) becomes

∫_{0}^{∞} |g(α)|w(α) dα ≤ ∫_{0}^{∞} |f(β)|β w(β) dβ
≤ ∫_{0}^{∞} |f(β)| w̃(β) dβ.

Finally, we need the case when z_{0} = ∞ where we need to trade off some smoothness of L^{−1}(F) to “divide out” a single zero of F at infinity.

Theorem A.7. Let F ∈ 𝔄_{n,w} with n ≥ 1 and suppose that F (∞) = 0. Then G(z) = zF (z) lies in 𝔄_{n−1,w}.

Proof. By an integration by parts, if α ∈ D(A(w)) and f ∈ C_{n,w}, then

z ∫_{0}^{∞} e^{αz}f(α) dα = f(0) − ∫_{0}^{∞} e^{αz}f′(α) dα.

The final result we need is a simple consequence of the earlier theorems. Note that if w is a weight and w̃ is given by (A.16) and F ∈ 𝔄_{n,w̃}, then F is C^{1} on the finite part of ∂D(A(w)). So it makes sense to talk about a simple zero (i.e., F (z_{0}) = 0, F′(z_{0}) ≠ 0) for z_{0} ∈ ∂D(A(w)).

Theorem A.8. Let F ∈ 𝔄_{n,w} with n ≥ 1. Suppose that the only zeros of F on D(A(w)) are at z_{1}, …, z_{k} all in D(A(w))^{int} and are simple. Then there is G ∈ 𝔄_{n,w} and a_{1}, …, a_{k} ∈ ℂ so that

F (z)^{−1} = ∑_{ℓ=1}^{k} \frac{a_{ℓ}}{z − z_{ℓ}} + G(z).  (A.17)

Equivalently for z > max(Re(z_{ℓ})),

F (z)^{−1} = F (∞)^{−1} + ∫_{0}^{∞} e^{−αz}h(α) dα,  (A.18)

where

h(α) = − ∑_{ℓ=1}^{k} a_{ℓ}e^{αz_{ℓ}} + g(α),  (A.19)

where g ∈ C_{n,w}. The result remains true if a single z_{ℓ} has Re z_{ℓ} = A(w) so long as F ∈ 𝔄_{n, ˜w̃} with ˜ w̃ given by

˜w̃(α) = (1 + |α|)^{2}w(α).

We will need

Lemma A.9. Let F ∈ 𝔄_{n,w} and z_{0} ∈ D(A(w))^{int} with F (z_{0}) ≠ 0. Then

\frac{F (z)}{z − z_{0}} = \frac{F (z_{0})}{z − z_{0}} + G(z)

with G ∈ 𝔄_{n+1,w}. If Re z_{0} = A(w) and F ∈ 𝔄_{n,w̃} with w̃ given by (A.16), then G ∈ 𝔄_{n+1,w}.

Proof of Lemma A.9. Write F (z) = H(z) + F (z_{0}), where H(z_{0}) = 0 and use Theorems A.4 and A.6.

Proof of Theorem A.8. By repeated use of Theorem A.4 and Corollary A.5, we can write

F (z) = H(z) ∏_{j=1}^{m} (\frac{z − z_{j}}{z − z_{0}} ) ,

where H is everywhere non-vanishing (it is because of the z −z_{0} factors that G(∞) ≠ 0). If all z_{j} lie in D(A(w))^{int}, then H ∈ 𝔄_{n+1,w}. If one z_{j} is in D(A(w)) and F ∈ 𝔄_{n, ˜w̃}, then H ∈ 𝔄_{n, ˜w̃}.

As a result,

F (z)^{−1} = H(z)^{−1} ∏_{j=1}^{m} [1 + \frac{z_{0} − z_{j}}{z − z_{j}} ] ,

where H(z)^{−1} is in 𝔄_{n,w} (resp. by 𝔄_{n,w̃}) by Theorem A.2.

By Lemma A.9,

(1 + \frac{z_{0} − z_{1}}{z − z_{1}} ) H(z) = \frac{c_{1}}{z − z_{1}} + H_{1}(z),

where H_{1} ∈ 𝔄_{n,w} (resp. 𝔄_{n,w̃}). Since

(1 + \frac{d_{1}}{z − z_{2}} ) (\frac{c_{1}}{z − z_{1}} ) = \frac{c_{2}}{z − z_{1}} + \frac{d_{2}}{z − z_{2}}

for suitable c_{2}, d_{2}, we obtain the result inductively.

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