A new approach to inverse spectral theory, I. Fundamental formalism

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Annals of Mathematics,150(1999), 1029–1057

A new approach to inverse spectral theory, I. Fundamental formalism

ByBarry Simon

Abstract

We present a new approach (distinct from Gel⁰fand-Levitan) to the theorem of Borg-Marchenko that them-function (equivalently, spectral measure) for a finite interval or half-line Schr¨odinger operator determines the potential. Our approach is an analog of the continued fraction approach for the moment problem. We prove there is a representation for the m-function m(−κ²) = −κ−R_b

0 A(α)e⁻^2ακdα+O(e⁻^(2b⁻^ε)κ). A on [0, a] is a function ofq on [0, a] and vice-versa. A key role is played by a differential equation that Aobeys after allowing x-dependence:

∂A

∂x = ∂A

∂α + Z _α

0

A(β, x)A(α−β, x)dβ.

Among our new results are necessary and sufficient conditions on the m- functions for potentialsq1 andq2 forq1 to equalq2 on [0, a].

1. Introduction

Inverse spectral methods have been actively studied in the past years both via their relevance in a variety of applications and their connection to the KdV equation. A major role is played by the Gel⁰fand-Levitan equations.

Our goal in this paper is to present a new approach to their basic results that we expect will lead to resolution of some of the remaining open questions in one-dimensional inverse spectral theory. We will introduce a new basic object (see (1.24) below), the remarkable equation, (1.28), it obeys and illustrate with several new results.

To present these new results, we will first describe the problems we dis- cuss. We will consider differential operators on eitherL²(0, b) with b < ∞ or L²(0,∞) of the form

(1.1) − d²

dx² +q(x).

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Ifbis finite, we suppose

(1.2) β1 ≡

Z b 0

|q(x)|dx <∞ and place a boundary condition

(1.3) u⁰(b) +hu(b) = 0,

whereh∈R∪ {∞}withh=∞shorthand for the Dirichlet conditionu(b) = 0.

Ifb=∞, we suppose (1.4)

Z y+1 y

|q(x)|dx <∞ for all y and

(1.5) β2≡sup

y>0

Z _y+1

y

max(q(x),0)dx <∞.

Under condition (1.5), it is known that (1.1) is the limit point at infinity [15].

In either case, for each z ∈ C\[β,∞) with −β sufficiently large, there is a unique solution (up to an overall constant), u(x, z), of −u⁰⁰ +qu = zu which obeys (1.3) at bif b <∞ or which isL² at ∞ ifb =∞. The principal m-function m(z) is defined by

(1.6) m(z) = u⁰(0, z)

u(0, z) .

We will sometimes need to indicate the q-dependence explicitly and write m(z;q). If b < ∞, “q” is intended to include all of q on (0, b), b, and the value of h.

If we replace bby b1 =b−x0 withx0∈(0, b) and letq(s) =q(x0+s) for s∈(0, b1), we get a newm-function we will denote bym(z, x0). It is given by

(1.7) m(z, x) = u⁰(x, z)

u(x, z) . m(z, x) obeys the Riccati equation

(1.8) dm

dx =q(x)−z−m²(z, x).

Obviously, m(z, x) only depends on q on (x, b) (and on h if b < ∞). A basic result of the inverse theory says that the converse is true:

Theorem1.1 (Borg [3], Marchenko [12]). m determinesq. Explicitly,if q1, q2 are two potentials and m1(z) =m2(z), thenq1 ≡q2 (including h1 =h2).

We will improve this as follows:

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NEW APPROACH TO INVERSE SPECTRAL THEORY, I 1031 Theorem 1.2. If (q1, b1, h1), (q2, b1, h2) are two potentials and a <

min(b1, b2), and if

(1.9) q1(x) =q2(x) on (0, a), then as κ→ ∞,

(1.10) m1(−κ²)−m2(−κ²) = ˜O(e⁻^2κa).

Conversely, if(1.10) holds, then(1.9)holds.

In (1.10), we use the symbol ˜O defined by f = ˜O(g) as x → x0 (where limx→x0g(x) = 0) if and only if limx→x0

|f(x)|

|g(x)|¹^−ε = 0 for all ε >0.

From a results point of view, this local version of the Borg-Marchenko uniqueness theorem is our most significant new result, but a major thrust of this paper are the new methods. Theorem 1.2 says that q is determined by the asymptotics ofm(−κ²) asκ → ∞. We can also read off differences of the boundary condition from these asymptotics. We will also prove that

Theorem 1.3. Let (q1, b1, h1), (q2, b2, h2) be two potentials and suppose that

(1.11) b1 =b2≡b <∞, |h1|+|h2|<∞, q1(x) =q2(x) on (0, b).

Then

(1.12) lim

κ→∞e^2bκ|m1(−κ²)−m2(−κ²)|= 4(h1−h2).

Conversely, if (1.12)holds for some b <∞ with a limit in (0,∞), then (1.11) holds.

Remark. That (1.11) implies (1.12) is not so hard to see. It is the converse that is interesting.

To understand our new approach, it is useful to recall briefly the two approaches to the inverse problem for Jacobi matrices on`²({0,1,2, . . . ,}) [2], [8], [18]:

A=







b0 a0 0 0 . . . a0 b1 a1 0 . . . 0 a1 b2 a2 . . . . . . . . . . . . . . . . . .







withai >0. Here the m-function is just (δ0,(A−z)⁻¹δ0) = m(z) and, more generally, mn(z) = (δn,(A⁽ⁿ⁾−z)⁻¹δn) with A⁽ⁿ⁾ on `²({n, n+ 1, . . . ,}) ob- tained by truncating the first n rows and n columns of A. Here δn is the Kronecker vector, that is, the vector with 1 in slotnand 0 in other slots. The fundamental theorem in this case is that m(z) ≡ m0(z) determines the bn’s andan’s.

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mn(z) obeys an analog of the Riccati equation (1.8):

(1.13) a²_nmn+1(z) =bn−z− 1 mn(z).

One solution of the inverse problem is to turn (1.13) around to see that (1.14) mn(z)⁻¹ =−z+bn−a²_nmn+1(z)

which, first of all, implies that asz → ∞,mn(z) =−z⁻¹+O(z⁻²); so (1.14) implies

(1.15) mn(z)⁻¹=−z+bn+a²_nz⁻¹+O(z⁻²).

Thus, (1.15) for n = 0 yields b0 and a²₀ and so m1(z) by (1.13), and then an obvious induction yields successivebk,a²_k, and mk+1(z).

A second solution involves orthogonal polynomials. Let Pn(z) be the eigensolutions of the formal (A − z)Pn = 0 with boundary conditions P₋1(z) = 0,P0(z) = 1. Explicitly,

(1.16) Pn+1(z) =a⁻_n¹[(z−bn)Pn(z)]−an−1Pn−1. Letdρ(x) be the spectral measure forA and vectorδ0 so that

(1.17) m(z) =

Z dρ(x) x−z. Then one can show that

(1.18)

Z

Pn(x)Pm(x)dµ(x) =δnm, n, m= 0,1, . . . .

Thus, Pn(z) is a polynomial of degree nwith positive leading coefficients determined by (1.18). These orthonormal polynomials are determined via Gram-Schmidt from ρ and by (1.17) from m. Once one has the Pn, one can determine thea’s andb’s from the equation (1.16).

Of course, these approaches via the Riccati equation and orthogonal polynomials are not completely disjoint. The Riccati solution gives the an’s and bn’s as continued fractions. The connection between continued fractions and orthogonal polynomials goes back a hundred years to Stieltjes’ work on the moment problem [18].

The Gel⁰fand-Levitan-Marchenko [7], [11], [12], [13] approach to the continuum case is a direct analog of this orthogonal polynomial case. One looks at solutionsU(x, k) of

(1.19) −U⁰⁰+q(x)U =k²U(x)

obeying U(0) = 1, U⁰(0) =ik, and proves that they obey a representation (1.20) U(x, k) =e^ikx+

Z _x

−x

K(x, y)e^ikydy,

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NEW APPROACH TO INVERSE SPECTRAL THEORY, I 1033 the analog ofPn(z) =czⁿ+ lower order. One definess(x, k) = (2ik)⁻¹[U(x, k)

−U(x,−k)] which obeys (1.19) withs(0) = 0, s⁰(0) = 1.

The spectral measure dρ associated to m(z) by dρ(λ) = lim

ε↓0[(2π)⁻¹ Imm(λ+iε)dλ]

obeys (1.21)

Z

s(x, k)s(y, k)dρ(k²) =δ(x−y),

at least formally. (1.20) and (1.21) yield an integral equation forK depending only on dρ and then once one has K, one can find U and so q via (1.19) (or via another relation betweenK and q).

Our goal in this paper is to present a new approach to the continuum case, that is, an analog of the Riccati equation approach to the discrete inverse problem. The simple idea for this is attractive but has a difficulty to overcome.

m(z, x) determines q(x) at least if q is continuous by the known asymptotics ([4]):

(1.22) m(−κ², x) =−κ−q(x)

2κ +o(κ⁻¹).

We can therefore think of (1.8) withqdefined by (1.22) as an evolution equation for m. The idea is that using a suitable underlying space and uniqueness theorem for solutions of differential equations, (1.8) should uniquely determine mfor all positive x, and so q(x) by (1.22).

To understand the difficulty, consider a potential q(x) on the whole real line. There are then functions u_±(x, z) defined for z ∈ C\[β,∞) which are L² at ±∞ and two m-functions m_±(z, x) = ^u_u⁰^±^(x,z)

±(x,z). Both obey (1.8), yet m+(0, z) determines and is determined byq on (0,∞) while m₋(0, z) has the same relation to q on (−∞,0). Put differently, m+(0, z) determines m+(x, z) forx >0 but not at all for x <0. m₋ is the reverse. So uniqueness for (1.8) is one-sided and either side is possible! That this does not make the scheme hopeless is connected with the fact thatm₋ does not obey (1.22); rather (1.23) m₋(−κ², x) =κ+q(x)

2κ +o(κ⁻¹).

We will see the one-sidedness of the solubility is intimately connected with the sign of the leading ±κ term in (1.22) and (1.23).

The key object in this new approach is a function A(α) defined for α∈(0, b) related tom by

(1.24) m(−κ²) =−κ− Z a

0

A(α)e⁻^2ακdα+ ˜O(e⁻^2aκ)

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as κ → ∞. We have written A(α) as a function of a single variable but we will allow similar dependence on other variables. Since m(−κ², x) is also an m-function, (1.24) has an analog with a function A(α, x). We will also some- times consider theq-dependence explicitly, usingA(α, x;q) or forλreal and q fixedA(α, x;λ)≡A(α, x;λq). If we are interested inq-dependence but not x, we will sometimes useA(α;λ). The semicolon and context distinguish between A(α, x) and A(α;λ).

By uniqueness of inverse Laplace transforms (see Theorem A.2.2 in Ap- pendix 2), (1.24) andm near−∞ uniquely determineA(α).

Not only will (1.24) hold but, in a sense,A(α) is close toq(α). Explicitly, in Section 3 we will prove that

Theorem 1.4. Let m be the m-function of the potential q. Then there is a function A(α) ∈ L¹(0, b) if b < ∞ and A(α) ∈ L¹(0, a) for all a < ∞ if b=∞ so that (1.24) holds for any a≤b with a <∞. A(α) only depends on q(y) for y ∈[0, α]. Moreover, A(α) =q(α) +E(α) where E(α) is continuous and obeys

(1.25) |E(α)| ≤ µZ α

0

|q(y)|dy

¶2

exp µ

α Z _α

0

|q(y)|dy

¶ .

Restoring the x-dependence, we see that A(α, x) = q(α+x) +E(α, x) where

limα↓0 sup

0≤x≤a|E(α, x)|= 0 for anya >0; so

(1.26) lim

α↓0A(α, x) =q(x),

where this holds in general inL¹ sense. Ifq is continuous, (1.26) holds point- wise. In general, (1.26) will hold at any point of right Lebesgue continuity ofq.

BecauseE is continuous,Adetermines any discontinuities or singularities of q. More is true. Ifq is C^k, thenE is C^k+2 in α, and so A determines k^th order kinks inq. Much more is true. In Section 7, we will prove

Theorem 1.5. q on [0, a] is only a function of A on[0, a]. Explicitly,if q1, q2 are two potentials, let A1, A2 be their A-functions. If a < b1, a < b2, andA1(α) =A2(α) for α ∈[0, a], thenq1(x) =q2(x) for x∈[0, a].

Theorems 1.4 and 1.5 immediately imply Theorem 1.2. For by Theo- rem A.2.2, (1.10) is equivalent toA1(α) =A2(α) for α∈[0, a]. Theorems 1.4 and 1.5 say this holds if and only if q1(x) =q2(x) for x∈[0, a].

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NEW APPROACH TO INVERSE SPECTRAL THEORY, I 1035 As noted, the singularities ofq come from singularities ofA. A boundary condition is a kind of singularity, so one might hope that boundary conditions correspond to very singularA. In essence, we will see that this is the case — there are delta-function and delta-prime singularities atα = b. Explicitly, in Section 5, we will prove that

Theorem 1.6. Let m be the m-function for a potential q with b < ∞. Then fora <2b,

(1.27) m(−κ²) =−κ− Z _a

0

A(α)e⁻^2ακdα−A1κe⁻^2κb−B1e⁻^2κb+ ˜O(e⁻^2aκ), where

(a) If h=∞, thenA1 = 2, B1=−2R_b

0 q(y)dy (b) If |h|<∞, thenA1 =−2, B1= 2[2h+R_b

0 q(y)dy].

As we will see in Section 5, this implies Theorem 1.3.

The reconstruction theorem, Theorem 1.5, depends on the differential equation that A(α, x) obeys. Remarkably, q drops out of the translation of (1.8) to the equation forA:

(1.28) ∂A(α, x)

∂x = ∂A(α, x)

∂α +

Z _α

0

A(β, x)A(α−β, x)dβ.

Ifq isC¹, the equation holds in the classical sense. For generalq, it holds in a variety of weaker senses. Either way, A(α,0) for α ∈ [0, a] determines A(α, x) for all x, αwithα >0 and 0< x+α < a. (1.26) then determinesq(x) forx∈[0, a). That is the essence of where uniqueness comes from.

Here is a summary of the rest of this paper. In Section 2, we start the proof of Theorem 1.4 by considering b =∞ and q ∈ L¹(0,∞). In that case, we prove a version of (1.24) with no error; namely, A(α) is defined on (0,∞) obeying

|A(α)−q(α)| ≤ kqk²1exp(αkqk1) and ifκ > ¹₂kqk1, then

(1.29) m(−κ²) =−κ−

Z _∞

0

A(α)e⁻^2ακdα.

In Section 3, we use this and localization estimates from Appendix 1 to prove Theorem 1.4 in general. Section 4 is an aside to study implications of (1.24) for asymptotic expansions. In particular, we will see that

(1.30) m(−κ²) =−κ− Z a

0

q(α)e⁻^2ακdα+o(κ⁻¹),

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which is essentially a result of Atkinson [1]. In Section 5, we turn to proofs of Theorems 1.6 and 1.3. Indeed, we will prove an analog of (1.27) for any a <∞. Ifa < nb, then there are terms P_n

m=1(Amκe⁻^2mκb+Bme⁻^2mκb) with explicitAm and Bm.

In Section 6, we prove (1.28), the evolution equation for A. In Section 7, we prove the fundamental uniqueness result, Theorem 1.5. Section 8 includes various comments including the relation to the Gel⁰fand-Levitan approach and a discussion of further questions raised by this approach.

I thank P. Deift, I. Gel⁰fand, R. Killip, and especially F. Gesztesy, for useful comments, and M. Ben-Artzi for the hospitality of Hebrew University where part of this work was done.

2. Existence of A: The L¹ case

In this section, we prove that whenq ∈L¹, then (1.29), which is a strong version of (1.24), holds. Indeed, we will prove

Theorem 2.1. Let q ∈ L¹(0,∞). Then there exists a function A(α) on (0,∞) with A−q continuous,obeying

(2.1) |A(α)−q(α)| ≤Q(α)²exp(αQ(α)), where

(2.2) Q(α)≡

Z _α

0 |q(y)|dy;

thus if κ > ¹₂kqk1, then

(2.3) m(−κ²) =−κ−

Z _∞

0

A(α)e⁻^2ακdα.

Moreover,if q,q˜are both in L¹,then

(2.4) |A(α;q)−A(α; ˜q)| ≤ kq−qk˜ 1[Q(α) + ˜Q(α)] exp(α[Q(α) + ˜Q(α)]).

We begin the proof with several remarks. First, since m(−κ²) is analytic inC\[β,∞), we need only prove (2.3) for all sufficiently largeκ. Second, since m(−κ²;qn)→ m(−κ²;q) as n→ ∞if kqn−qk1 →0, we can use (2.4) to see that it suffices to prove the theorem ifq is a continuous function of compact support, which we do henceforth. So suppose q is continuous and supported in [0, B].

We will prove the following:

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NEW APPROACH TO INVERSE SPECTRAL THEORY, I 1037 Lemma 2.2. Let q be a continuous function supported on [0, B]. For λ ∈ R, let m(z;λ) be the m-function for λq. Then for any z ∈ C with dist(z,[0,∞))> λkqk∞,

(2.5a) m(z;λ) =−κ−

X∞ n=1

Mn(z;q)λⁿ, where forκ >0,

(2.5b) Mn(−κ²;q) = Z _nB

0

An(α)e⁻^2καdα, where

(2.6) A1(α) =q(α)

and for n≥2, An(α) is a continuous function obeying (2.7) |An(α)| ≤Q(α)ⁿ αⁿ⁻²

(n−2)!. Moreover,if q˜is a second such potential andn≥2, (2.8) |An(α;q)−An(α; ˜q)| ≤(Q(α)+ ˜Q(α))ⁿ⁻¹

· Z _α

0

|q(y)−q(y)˜ |dy

¸ αⁿ⁻² (n−2)!. Proof of Theorem 2.1given Lemma 2.2. By (2.7),

Z _∞

0

X∞ n=2

|An(α)|e⁻^2καdα <∞

ifκ > ¹₂kqk1. Thus in (2.5a) forλ= 1, we can interchange the sum and integral to get the representation (2.3). (2.7) then implies (2.1) and (2.8) implies (2.4).

Proof of Lemma 2.2. LetHλ be −_dx^d²2 +λq(x) onL²(0,∞) withu(0) = 0 boundary conditions at 0. Then k(H0−z)⁻¹k= dist(z,[0,∞))⁻¹. So, in the sense ofL² operators, if dist(z,[0,∞))> λkqk∞, the expansion

(2.9) (Hλ−z)⁻¹ = X∞ n=0

(−1)ⁿ(H0−z)⁻¹[λq(H0−z)⁻¹]ⁿ is absolutely convergent.

As is well known, Gλ(x, y;z), the integral kernel of (Hλ −z)⁻¹, can be written down in terms of the solutionuwhich isL² at infinity, and the solution wof

(2.10) −w⁰⁰+qw=zw

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obeying w(0) = 0, w⁰(0) = 1

(2.11) Gλ(x, y;z) =w(min(x, y))u(max(x, y)) u(0) . In particular,

(2.12) m(z) = lim

x<y y↓0

∂²G

∂x∂y. From this and (2.9), we see that (using ^∂G_∂x⁰(x, y)¯¯¯

x=0=e⁻^κy) m(−κ²;λ) =−κ−λ

Z

e⁻^2κyq(y)dy+λ²hϕκ,(Hλ+κ²)⁻¹ϕκi,

whereϕκ(y) =q(y)e⁻^κy. Sinceϕκ ∈L², we can use the convergent expansion (2.9) and so conclude that (2.5a) holds with (forn≥2)

Mn(−κ²;q) = (−1)ⁿ⁻¹ Z

e⁻^κx¹q(x1)G0(x1, x2)q(x2) . . . G0(xn−1, xn)q(xn)e⁻^κxⁿdx1. . . dxn. (2.13)

Now use the following representation for G0: G0(x, y;−κ²) = sinh(κmin(x, y))

κ e⁻^κ^max(x,y) (2.14)

= 1 2

Z _x+y

|x−y|e⁻^`κd`

to write (2.15)

Mn(−κ²;q)

= (−1)ⁿ⁻¹ 2ⁿ⁻¹

Z

Rn

q(x1). . . q(xn)e⁻^2α(x¹^,xⁿ^,`¹^,...,`ⁿ⁻¹^)κdx1. . . dxnd`1. . . d`n−1, whereα is shorthand for the linear function

(2.16) α= 1

2 µ

x1+xn+

n−1

X

j=1

`j

¶

andRn is the region

Rn={(x1, . . . , xn,`1, . . . , `n−1)∈R²ⁿ⁻¹ |0≤xi ≤B fori= 1, . . . , n;

|xi−xi+1| ≤`i≤xi+xi−1 fori−1, . . . , n−1}.

In the regionRn, notice that α≤ 1

2 µ

x1+xn+

n−1

X

j=1

(xj+xj+1)

¶

= Xn j=1

xj ≤nB.

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NEW APPROACH TO INVERSE SPECTRAL THEORY, I 1039 Change variables by replacing `n−1 by α using the linear transformation (2.16) and use`n−1 for the linear function

(2.17) `n−1(x1, xn, `1, . . . , `n−2, α) = 2α−x1−xn−

n−2

X

j=1

`j. Thus, (2.5b) holds where

(2.18) An(α) = (−1)ⁿ⁻¹ 2ⁿ⁻²

Z

R_n(α)

q(x1). . . q(xn)dx1. . . dxnd`1. . . d`n−2. 2ⁿ⁻¹ has become 2ⁿ⁻² because of the Jacobian of the transition from `n−1

toα. Rn(α) is the region (2.19)

Rn(α) ={(x1, . . . , xn, `1, . . . , `n−2)∈R²ⁿ⁻² |0≤xi ≤B fori= 1, . . . , n;

|xi−xi+1| ≤`i ≤xi+xi+1 fori= 1, . . . , n−2;

|xn−1+xn| ≤`n−1(x1, . . . , xn, `1, . . . , `n−2, α)≤xn−1+xn} with`n−1 the functional given by (2.17).

We claim that Rn(α)⊂R˜n(α) (2.20)

=

½

(x1, . . . , xn, `1, . . . , `n−2)∈R²ⁿ⁻²¯¯

¯¯0≤xi≤α;`i≥0;

n−2

X

i=1

`i≤2α

¾ . Accepting (2.20) for a moment, we note by (2.18) that

|An(α)| ≤ 1 2ⁿ⁻²

Z

R˜_n(α)|q(x1)|. . .|q(xn)|dx1. . . d`n−2

= µZ _α

0

|q(x)|dx

¶n

αⁿ⁻² (n−2)!

sinceR_P

y_i=b;y_i≥0dy1. . . dyn= ^b_n!ⁿ by a simple induction. This is just (2.7).

To prove (2.8), we note that

|An(α;q)−An(α,q)˜|

≤2⁻ⁿ⁻² Z

R˜_n(α)|q(x1). . . q(xn)−q(x˜ 1). . .q(x˜ n)|dx1. . . d`n−2

≤ αⁿ⁻² (n−2)!

n−1

X

j=0

Q(α)^j

·Z _α

0

|q(y)−q(y)˜ |dy

¸

Q(α)˜ ⁿ⁻^j⁻¹. SinceP_m

j=0a^jb^m⁻^j ≤P_m

j=0

¡_m

j

¢a^jb^m⁻^j = (a+b)^m, (2.8) holds.

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Thus, we need only prove (2.20). Suppose (x1, . . . , xn, `1, . . . , `n−2) ∈ Rn(α). Then

2xm ≤ |x1−xm|+|xn−xm|+x1+xn

≤x1+xn+

n−1

X

j=1

|xj+1−xj|

≤x1+xn+

n−2

X

j=1

`j+`n−1(x1, . . . , xn, `1, . . . , `n−2;α) = 2α

so 0≤xj ≤α, proving that part of the condition (x1, `n−2)⊂R˜n(α). For the second part, note that

n−2

X

j=1

`j = 2α−x1−xn−`n−1(x1, . . . , xn, `1, . . . , `n−2)≤2α sincex1,xn, and `n−2 are nonnegative onRn(α).

We want to say more about the smoothness of the functions An(α) and An(α, x) defined forx≥0 and n≥2 by

(2.21)

An(α, x) = (−1)ⁿ⁻¹ 2ⁿ⁻²

Z

R_n(α)

q(x+x1). . . q(x+xn)dx1. . . dxnd`1. . . d`n−2

so that A(α, x) = P_∞

n=0An(α, x) is the A-function associated to m(−κ², x).

We begin withα smoothness for fixed x.

Proposition2.3. An(α, x) is a Cⁿ⁻²-function inα and obeys forn≥3 (2.22) ¯¯

¯¯d^jAn(α) dα^j

¯¯¯¯≤ 1

(n−2−j)!αⁿ⁻²⁻^jQ(α)ⁿ; j= 1, . . . , n−2.

Proof. Write An(α) = (−1)ⁿ⁻¹

2ⁿ⁻¹ Z

Rn

q(x1). . . q(xn)δ µ

2α−x1−xn−

nX−1 m=1

`i

¶ dx1

. . . dxnd`j. . . d`n−1. Thus, formally,

d^jAn(α)

dα^j = (−1)ⁿ⁻¹2^j 2ⁿ⁻²

Z

R_n

q(x1) (2.23)

. . . q(xn)δ^(j) µ

2α−x1−xn−

n−1

X

m=1

`i

¶

dx1. . . d`n−1.

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NEW APPROACH TO INVERSE SPECTRAL THEORY, I 1041 Since j+ 1 ≤ n−1, we can successively integrate out `n−1, `n−2, . . . , `n−j−1

using (2.24)

Z _b

a

δ^j(c−`)d`=δ^j⁻¹(c−a)−δ^j⁻¹(c−b) and

(2.25)

Z b a

δ(c−`)d`=χ(a,b)(c).

Then we estimate each of the resulting 2^j terms as in the previous lemma, getting

¯¯¯¯d^jAn(α) dα^j

¯¯¯¯≤ 2^j

2ⁿ⁻²Q(α)ⁿ (2α)ⁿ⁻^j⁻² (n−j−2)!

which is (2.22).

(2.24), (2.25), while formal, are a way of bookkeeping for legitimate move- ment of hyperplanes. In (2.25), there is a singularity atc =aand c =b, but since we are integrating in further variables, these are irrelevant.

Proposition 2.4. If q isC^m, thenAn(α) isC^m+(2n⁻²⁾.

Proof. Write Rn as n! terms with orderings xπ(1) < · · · < xπ(n). For j0 = 2n−2, we integrate out all 2n−1, ` and x variables. We get a formula for ^d^j⁰_dα^Aⁿ_j0^(α) as a sum of products of q’s evaluated at rational multiples of α.

We can then take madditional derivatives.

Theorem2.5. If q is C^m and in L¹(0,∞), then A(α) is C^m and A(α)

−q(α) is C^m+2.

Proof. By (2.2), we can sum the terms in the series for ^d_dα^j^A_j and ^d^j^(A_dα⁻_j^q) forj= 0,1, . . . , m and j= 0,1, . . . , m−2, respectively. With this bound and the fundamental theorem of calculus, one can prove the stated regularity.

Now we can turn to x-dependence.

Lemma2.6. Ifq isC^k and of compact support,thenAn(α, x)for α fixed isC^k in x,and for n≥2, j= 1, . . . , k,

(2.26) ¯¯

¯¯d^jAn(α, x) dx^j

¯¯¯¯≤Q(α)^max(0,n⁻^j)[Pj(α)]^min(j,n) αⁿ⁻² (n−2)!, where

Pj(α) = Z _α

0

Xj m=0

¯¯¯¯d^mq dx^m (y)¯¯

¯¯ dy.

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Proof. In (2.21), we can take derivatives with respect tox. We get a sum of terms with derivatives on eachq, and using values on these terms and the argument in the proof of Lemma 2.2, we obtain (2.26).

Theorem 2.7. If q is C^k and of compact support, then A(α, x) for α fixed is C^k in x and

d^jm

dx^j (−κ², x) =− Z _∞

0

∂^jA

∂x^j (α, x)e⁻^2ακdα for κ large and j = 1,2, . . . , k.

Proof. This follows from the estimates in Lemma 2.6 and Theorem 2.1.

3. Existence of A: General case

By combining Theorem 2.1 and Theorem A.1.1, we immediately have Theorem 3.1. Let b < ∞, q ∈ L¹(0, b), and h ∈ R∪ {∞} or else let b = ∞ and let q obey (1.4), (1.5). Fix a < b. Then, there exists a function A(α) onL¹(0, a) obeying

(3.1) |A(α)−q(α)| ≤Q(α) exp(αQ(α)), where

(3.2) Q(α)≡

Z α 0

|q(y)|dy so that as κ→ ∞,

(3.3) m(−κ²) =−κ− Z _a

0

A(α)e⁻^2ακdα+ ˜O(e⁻^2aκ).

Moreover,A(α) on [0, a]is only a function of q on[0, a].

Proof. Let ˜b = ∞ and ˜q(x) = q(x) for x ∈ [0, a] and ˜q(x) = 0 for x > a. By Theorem A.1.1, m −m˜ = ˜O(e⁻^2aκ), and by Theorem 2.1, ˜m has a representation of the form (3.3).

4. Asymptotic formula

While our interest in the representation (1.24) is primarily for inverse theory and, in a sense, it provides an extremely complete form of asymptotics, the formula is also useful to recover and extend results of others on more conventional asymptotics.

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NEW APPROACH TO INVERSE SPECTRAL THEORY, I 1043 In this section, we will explain this theme. We begin with a result related to Atkinson [1] (who extended Everitt [5]).

Theorem 4.1. For any q (obeying (1.2)–(1.5)), we have that (4.1) m(−κ²) =−κ−

Z b 0

q(x)e⁻^2xκdx+o(κ⁻¹).

Remarks. 1. Atkinson’s “m” is the negative inverse of ourmand he uses k=iκ, and so his formula reads ((4.3) in [1])

mAtk(k²) =ik⁻¹+k⁻² Z _b

0

e^2ikxq(x)dx+o(|k|⁻³).

2. Atkinson’s result is stronger in that he allows cases where q is not bounded below (and so he takes|z| → ∞staying away from the negative real axis also). [10] will extend (4.1) to some such situations.

3. Atkinson’s method breaks down on the realx axis where our estimates hold, but one could use Phragm´en-Lindel¨of methods and Atkinson’s results to prove Theorem 4.1.

Proof. By Theorem 3.1, (A−q)→0 asα↓0 soR_a

0 e⁻^2aκ(A(α)−q(α))dα

=o(κ⁻¹). Thus, (3.3) implies (4.1).

Corollary 4.2.

m(−κ²) =−κ+o(1).

Proof. Sinceq ∈L¹, dominated convergence implies thatRb

0 q(x)e⁻^2κxdx

=o(1).

Corollary4.3. If lim_x_↓₀q(x) =a(indeed,if ¹_sRs

0 q(x)dx→aass↓0), then

m(−κ²) =−κ−a

2κ⁻¹+o(κ⁻¹).

Corollary 4.4. If q(x) =cx⁻^α+o(x⁻^α) for 0< α <1,then m(−κ²) =−κ−c[2^a⁻¹Γ(1−α)]κ^α⁻¹+o(κ^α⁻¹).

We can also recover the result of Danielyan and Levitan [4]:

Theorem4.5. Let q(x)∈Cⁿ[0, δ) for someδ >0. Then as κ→ ∞,for suitableβ0, . . . , βn, we have that

(4.2) m(−κ²) =−κ− Xn m=0

βjκ⁻^j⁻¹+O(κ⁻ⁿ⁻¹).

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Remarks. 1. Ourm is the negative inverse of theirm.

2. Our proof does not require that q is Cⁿ. It suffices that q(x) has an asymptotic seriesP_n

m=0amx^m+o(xⁿ) as x↓0.

Proof. By Theorems 3.1 and 2.5, A(α) is Cⁿ on [0, δ). It follows that A(α) =P_n

m=0bjα^j+o(α^j). Since R_δ

0 α^je⁻^2ακdα=κ⁻^j⁻¹2⁻^j⁻¹j! + ˜O(e⁻^2δκ), we have (4.2) βj = 2^j⁻¹j!bj = 2^j⁻¹^∂_∂α^j^A_j(α = 0).

Later we will prove that Aobeys (1.28). This immediately yields a recur- sion formula forβj(x), viz.:

βj+1(x) = 1 2

∂βj

∂x +1 2

Xj

`=0

β`(x)βj−`(x), j≥0 β0(x) = 1

2q(x);