ON THE DETERMINATION OF A

POTENTIAL FROM THREE SPECTRA

Fritz Gesztesy^{1} and Barry Simon^{2}

August 19, 1997

Dedicated to M.S. Birman on the occasion of his seventieth birthday

Abstract. We prove that under suitable circumstances, the spectra of a Schrödinger operator on the three intervals [0, 1], [0, a], and [a, 1] for some a ∈ (0, 1) uniquely determine the potential q on [0, 1].

§1. Introduction

This is a paper in our series [2,4,5,6] on the use of Weyl-Titchmarsh m-function methods to obtain information on what spectral information uniquely determines the potential q in a one-dimensional Schrödinger operator −\frac{d^{2}}{dx^{2}} + q. Typical of our results is:

Theorem 1. Fix c, d ∈ ℝ with c < d and q ∈ L^{1}((c, d)) real-valued. Let S(c, d; q) denote the set of eigenvalues of −\frac{d^{2}}{dx^{2}} + q on L^{2}((c, d)) with the boundary conditions u(c) = u(d) = 0. Suppose q_{1}, q_{2} ∈ L^{1}((0, 1)) are real-valued and there is some a ∈ (0, 1) so that

(i) S(0, 1; q_{1}) = S(0, 1; q_{2}), S(0, a; q_{1}) = S(0, a; q_{2}), and S(a, 1; q_{1}) = S(a, 1; q_{2}). (ii) The sets S(0, 1; q_{1}), S(0, a; q_{1}), and S(a, 1; q_{1}) are pairwise disjoint.

1991 Mathematics Subject Classification. Primary 34A55, 34B20; Secondary 34L05, 34L40. Key words and phrases. Inverse spectral theory, Schrödinger operators, Weyl-Titchmarsh m-functions. 1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: fritz@

math.missouri.edu Partially supported by the National Science Foundation under Grant No. DMS-9623121. 2 Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena,

CA 91125, USA. E-mail: bsimon@caltech.edu

To appear in the Birman Birthday Volume in Advances in Mathematical Sciences, V. Buslaev and M. Solomyak (eds.), Amer. Math. Soc., Providence, RI.

Typeset by AMS-TEX

Then q_{1} = q_{2} a.e. on [0, 1].

Our immediate motivation for this result is a recent preprint of Pivovarchik [13], who stated this result for a = \frac{1}{2} without condition (ii). As we will see in Section 5, there are counterexamples if (ii) fails. After we pointed out the relevance of condition (ii) to Pivovarchik, he provided a corrected version. One of our goals here is to show that the methods of [2,5,6] provide a natural way to understand and extend this result.

There is a second motivation for this work. While not stated in this language, we actually considered a problem very close to a finite-difference analog of Theorem 1 in [5]. There we considered a tridiagonal Jacobi matrix in ℂ^{N}

A = ( \begin{matrix}b_{1} & a_{1} & 0 & … \\ a_{1} & b_{2} & a_{2} & … \\ … & … & … & … \\ … & … & a_{N−1} & b_{N} \end{matrix} ) ,

with a_{k} > 0, k = 1, …, N − 1. Let A^{[i,j]} be the submatrix of A obtained by keeping rows and columns i, i + 1, …, j − 1, j. In [5] we considered to what extent A is determined by g(z, k), the kk matrix element of (A − z)^{−1} (for all z ∈ ℂ\spec(A)). We found that generically there were \binom{N−1}{k−1} possible A’s consistent with a given g(z, k).

The proof of this fact depends on the argument that looked at the eigenvalues of A^{[1,k−1]} and A^{[k+1,N]}. The function g(z, k) determined the union of these sets. Then \binom{N−1}{k−1} possible values depended on the choice of which were actually eigenvalues of A^{[1,k−1]} and which of A^{[k+1,N]}. If one a priori knows which are which (the hypothesis of Theorem 1), one has uniqueness.

The non-generic case in [5] occurs precisely when A^{[1,k]} and A^{[k+1,N]} share an eigenvalue, in which case there is a manifold of possible A’s consistent with g(z, k).

In a sense, Theorem 1 can be thought of as a continuum analog of a part of the result in [5].

We actually prove a more general result than Theorem 1. Let h_{c}, h_{d} ∈ ℝ ∪ {∞}. We let H(c, d; h_{c}, h_{d}; q) be the operator −\frac{d^{2}}{dx^{2}} + q on L^{2}((c, d)) with boundary conditions

u′(c) + h_{c}u(c) = 0, u′(d) + h_{d}u(d) = 0,

where h_{x_{0}} = ∞ is a shorthand notation for the Dirichlet boundary condition at x = x_{0} (i.e., u(x_{0}) = 0). Let S(c, d; h_{c}, h_{d}; q) be the set of eigenvalues (i.e., the spectrum) of H(c, d; h_{c}, h_{d}; q).

We will prove

Theorem 2. Fix a ∈ (0, 1) and h_{0}, h_{1}, h_{a} ∈ ℝ ∪ {∞}. Suppose q_{1}, q_{2} ∈ L^{1}((0, 1)) are real-valued and

(i) S(0, 1; h_{0}, h_{1}; q_{1}) = S(0, 1; h_{0}, h_{1}; q_{2}), S(0, a; h_{0}, h_{a}; q_{1}) = S(0, a; h_{0}, h_{a}; q_{2}), and S(a, 1; h_{a}, h_{1}; q_{1}) = S(a, 1; h_{a}, h_{1}; q_{2}).

(ii) The sets S(0, 1; h_{0}, h_{1}; q_{1}), S(0, a; h_{0}, h_{a}; q_{k}), and S(a, 1; h_{a}, h_{1}; q_{k}) are pairwise disjoint.

Then q_{1} = q_{2} a.e. on [0, 1].

Remark. The proof actually shows that not only is q determined by S(0, 1), S(0, a; h_{a}), and S(a, 1; h_{a}), but so are h_{0} and h_{1}.

The structure of this paper is as follows: In Section 2, we prove several results which illustrate when Green’s functions are determined by zeros, poles, and residues. In Section 3, we prove Theorem 2 when h_{a} = ∞ (including Theorem 1); and in Section 4, we prove Theorem 2 when |h_{a}| < ∞. In Section 5, we discuss the case where condition (ii) fails. In Section 6, we consider some cases where q is defined on all of ℝ.

It is a great pleasure to dedicate this paper as a seventieth birthday present to M.S. Birman, whose work has long inspired us. In our use of Green’s functions and analytic function theory, the reader will see echoes of his influence.

We thank V. Pivovarchik for sending us his manuscript [13] prior to publication. F.G. is indebted to A. S. Kechris and C. W. Peck for a kind invitation to Caltech for a month during the summer of 1997. The extraordinary hospitality and support by the Department of Mathematics at Caltech are gratefully acknowledged. B.S. would like to thank M. Ben-Artzi for the hospitality of Hebrew University where some of this work was done.

§2. Some Uniqueness Theorems of Meromorphic Herglotz Functions

One could prove the basic result of this paper using the theorems in [2,6] on the determination of an entire function by its values on a set of suitable density. Instead we will use some alternative theorems that allow ready extension to q’s on all of ℝ, a typical one being

Theorem 2.1. Let 0 < w_{1} < z_{1} < w_{2} < z_{2} < ⋯ with lim_{n→∞} w_{n} = ∞. Then

g(z) = lim_{n→∞} ∏_{j=1}^{n}(1 − z/z_{j})/ ∏_{j=1}^{n}(1 − z/w_{j})

exists for any z in ℂ\{w_{j}}_{j=1}^{∞}, with convergence uniform on compact subsets of ℂ\{w_{j}}_{j=1}^{∞}. g(z) is a meromorphic function with

\frac{Im (g(z))}{Im (z)} > 0 for z ∈ ℂ \ ℝ  (2.1)

and hence a Herglotz function. Moreover, any meromorphic function f(z) satisfying (2.1) with zeros precisely at {z_{j}}_{j=1}^{∞} and poles precisely at {w_{j}}_{j=1}^{∞} is a positive multiple of g(z).

Remarks. 1. Theorems of this genre can be found in Levin [7].

2. This is a variant of the standard theorem on the convergence of alternating series.

3. One can easily accommodate situations where there are also zeros and poles alternating towards −∞.

4. Any meromorphic Herglotz function (i.e., any meromorphic function satisfying (2.1)) can be seen to satisfy f′(z) > 0 away from its polar singularities, so its zeros and poles are simple, its zeros and poles alternate, and residues at poles are negative. Thus Theorem 2.1 describes all meromorphic Herglotz functions which are positive on (−∞, w_{1}) for some w_{1} > 0.

Proof. Let g_{N}(z) = ∏_{j=1}^{N}(1 − z/z_{j})/ ∏_{j=1}^{N+1}(1 − z/w_{j}). Then g_{N} has simple poles at w_{1}, w_{2}, …, w_{N+1} and because of the alternating nature of the z_{j}’s and w_{j}’s, each residue

is negative. Since g_{N}(z) → 0 as |z| → ∞, it follows that g_{N}(z) = ∑_{j=1}^{N+1} \frac{α_{j}^{(N)}}{w_{j}−z} with

α_{j}^{(N)} > 0, j = 1, …, N + 1.

Thus, each g_{N} is a Herglotz function and so g_{N} maps ℂ\[0, ∞) to ℂ\(−∞, 0]. Let H be a biholomorphic map of ℂ\(−∞, 0] to the open unit disk (e.g., H(w) = \frac{\sqrt{w}−1}{\sqrt{w}+1}). By applying the Vitali convergence theorem (see, e.g., [15], Ch. 5) to H ∘ g_{N}, we see it suffices to show g_{N}(x) converges for each x ∈ (−∞, 0) to conclude that g_{N}(z) converges as N → ∞ for z ∈ ℂ\(0, ∞).

Since w_{j} < z_{j}, we have (1 − x/z_{j})/(1 − x/w_{j}) < 1, and since w_{j+1} > z_{j}, we have (1 − x/z_{j})/(1 − x/w_{j+1}) > 1 assuming x < 0. Thus g_{1}(x) < g_{2}(x) < ⋯ < g_{N}(x) < g_{N+1}(x) < 1, so lim_{N→∞} g_{N}(x) exists for x < 0.

Once we have convergence on ℂ\(0, ∞), it is easy to extend the argument to ℂ\{w_{j}}_{j=1}^{∞}. Finally, let f(z) be a Herglotz function with the stated zeros and poles. Then f(z)/g(z)

is an entire non-vanishing function, and on ℂ\[0, ∞), |Im (ln(f(z)/g(z)))| ≤ 2π since |Im (ln(f(z)))| ≤ π and |Im (ln(g(z)))| ≤ π on ℂ \ [0, ∞). It follows that f(z)/g(z) is constant. □

In exactly the same way one infers

Theorem 2.2. Let 0 < z_{1} < w_{1} < z_{2} < w_{2} < ⋯ with lim_{n→∞} w_{n} = ∞. Then

g(z) = lim_{n→∞} ∏_{j=1}^{n}(1 − z/z_{j})/ ∏_{j=1}^{n}(1 − z/w_{j})

exists for any z in ℂ\{w_{j}}_{j=1}^{∞} with convergence uniform on compact subsets of ℂ\{w_{j}}_{j=1}^{∞}. g(z) is a meromorphic function with \frac{Im (g(z))}{Im (z)} < 0 for z ∈ ℂ \ ℝ.

Moreover, any meromorphic function f(z) satisfying (2.1) with zeros precisely at {z_{j}}_{j=1}^{∞} and poles precisely at {w_{j}}_{j=1}^{∞} is a negative multiple of g(z).

We also have theorems on asymptotics, poles, and residues determining a meromorphic Herglotz function.

Theorem 2.3. Let f_{1}(z), f_{2}(z) be two meromorphic Herglotz functions with identical sets of poles and residues, respectively. If

f_{1}(ix) − f_{2}(ix) → 0 as x → ∞,  (2.2)

then f_{1} = f_{2}.

Proof. By the Herglotz representation theorem, if f(z) is a meromorphic Herglotz function with poles at {w_{j}}_{j=1}^{∞} in ℝ and residues −α_{k} < 0 at z = w_{k}, then for some constants A ≥ 0 and B ∈ ℝ,

f(z) = Az + B + ∑_{j=1}^{∞} α_{j}[\frac{1}{w_{j} − z} − \frac{w_{j}}{1 + w_{j}^{2}} ],

where the sum is absolutely convergent since ∑_{j=1}^{∞} \frac{α_{j}}{1+w_{j}^{2}} < ∞.

Thus f_{1}(z) − f_{2}(z) = Ãz − B̃ for some Ã, B̃ ∈ ℝ, and therefore, (2.2) implies Ã = B̃ = 0. □

In applications, either f_{1}(ix) and f_{2}(ix) are both o(1) as x → ∞ or else, f_{1}(ix) and f_{2}(ix) are both \sqrt{ix} + o(1) as x → ∞.

§3. The Case of a Dirichlet Boundary Condition h_{a} = ∞

We want to prove Theorem 2 when h_{a} = ∞. If h_{0} < ∞, let u_{−}(z, x; q) solve −u′′+qu = zu with boundary conditions u_{−}(z, 0; q) = 1, u′_{−}(z, 0; q) = −h_{0}. If h_{0} = ∞, let it satisfy u_{−}(z, 0; q) = 0, u′_{−}(z, 0; q) = 1. As is well known (see, e.g., [11], Ch. 1), u_{−} is an entire function of z. Similarly, u_{+} satisfies the h_{1} boundary condition at 1.

Let

W (z; q) = u′_{−}(z, x; q)u_{+}(z, x; q) − u_{−}(z, x; q)u′_{+}(z, x; q),

which is independent of x. The zeros of W are precisely the points w_{i} of S(0, 1; h_{0}, h_{1}; q), that is, the eigenvalues of H := H(0, 1; h_{0}, h_{1}; q).

Fix a ∈ (0, 1) and q. Let g(z) = G(z, a, a) be the Green’s function of H in L^{2}((0, 1)) at (a, a), that is, the integral kernel of (H − z)^{−1} at (a, a). (We also use the notation g(z; q) for g(z) whenever the dependence of g(z) on q needs to be underscored.) Then, by a standard formula for the Green’s function of H,

g(z; q) = \frac{u_{−}(z, a; q)u_{+}(z, a; q)}{W (z; q)} .  (3.1)

The zeros of u_{+}(z, a; q) are precisely the points of S(a, 1; h_{a} = ∞, h_{1}; q) and the zeros of u_{−}(z, a; q) are precisely the points of S(0, a; h_{0}, h_{a} = ∞; q). The hypothesis (ii) on disjointness of the S sets in Theorem 2 says that the poles of g(z) are precisely the points of S(0, 1), and the zeros, the points of S(0, a) ∪ S(a, 1). (If the sets are not disjoint, there are cancellations between zeros and poles.)

By Theorem 2.1 (adding a constant to q if need be, we can assume all poles and zeros are positive), the zeros and poles of g(z) and the known asymptotics g(−κ^{2}; q) = (2κ)^{−1} (1 + o(1)) as κ → ∞ determine g, that is, g(z; q_{1}) = g(z; q_{2}).

Next we use the m-functions m_{±} defined by m_{±}(z; q) = ±u′_{±}(z, a; q)/u_{±}(z, a; q). By (3.1),

g(z; q) = −\frac{1}{[m_{+}(z; q) + m_{−}(z; q)]} .  (3.2)

Moreover, the poles of m_{+} (resp. m_{−}) are precisely the points λ of S(a, 1; h_{a} = ∞, h_{1}; q) (resp. S(0, a; h_{0}, h_{a} = ∞; q)). And the residues of the poles are determined by g. Explicitly, if λ_{0} is a pole of m_{+}, by hypothesis (ii) in Theorem 2, it is not a pole of m_{−}, and so its residue is −1/\frac{∂g}{∂z} |_{z=λ_{0}}.

By Theorem 2.2 and the asymptotics m_{±}(−κ^{2}; q) = −κ + o(1) as κ → ∞, the poles and residues determine m_{±}; that is, m_{±}(z; q_{1}) = m_{±}(z; q_{2}).

Finally, the uniqueness result of Borg [1] and Marchenko [12] guarantees that m_{±}(z; q) uniquely determine g on [0, a] and [a, 1], so q_{1} = q_{2} a.e. on [0, 1].

§4. The Case h_{a} ∈ ℝ

The changes in the proof when |h_{a}| < ∞ are minimal. Define u_{±} as in the last section, but instead of (3.1), define

g(z; q) = \frac{[u′_{−}(z, a; q) + h_{a}u_{−}(z, a; q)][u′_{+}(z, a; q) + h_{a}u_{+}(z, a; q)]}{W (z; q)} .  (4.1)

Since W = (u′_{−} + h_{a}u_{−})u_{+} − u_{−}(u′_{+} + h_{a}u_{+}), (3.2) becomes

g(z; q) = \frac{1}{\frac{1}{m_{+}(z;q)+h_{a}} + \frac{1}{m_{−}(z;q)−h_{a}}} .  (4.2)

The spectra determine the zeros and poles of g which, together with the asymptotics g(−κ^{2}; q) = −\frac{1}{2}κ(1 + o(1)) as κ → ∞, determine g by Theorem 2.1 or 2.2.

By hypothesis (ii) of Theorem 2, the poles of (m_{±} ± h_{a})^{−1} are distinct and so their residues are determined by (4.2) and the knowledge of g. The poles and residues of −(m_{±} ± h_{a})^{−1} and the fact that |m_{±}(ix)| → ∞ as x → ∞ determine (m_{±} ± h_{a})^{−1} by Theorem 2.3. The Borg-Marchenko uniqueness theorem then completes the proof.

§5. Examples of Non-Uniqueness

Our goal here is to show that if condition (ii) fails, then the uniqueness result in Theorem 2 can also fail. We will take an extreme case where S(0, \frac{1}{2}) = S(\frac{1}{2}, 1) for simplicity; but we have no doubt that a single point in common suffices to construct counterexamples to the extension of Theorem 2 with (ii) absent. We note that S(0, 1) ∩ S(0, \frac{1}{2}) = S(0, 1) ∩ S(\frac{1}{2}, 1) = S(0, \frac{1}{2}) ∩ S(\frac{1}{2}, 1) so that if two S’s fail to be disjoint, each pair has non-zero intersection.

To begin we note

Lemma 5.1. Let f be a continuous map of Q := [0, 1] × [0, 1] to the unit circle. Then, there exists a pair of points p_{0}, p_{1} ∈ Q with p_{0} ≠ p_{1} and f(p_{0}) = f(p_{1}).

Proof. If f(0, 0) = f(1, 1), we have the required points. If not, reparametrize the circle so that f(0, 0) = 1, f(1, 1) = −1. Consider the images f(γ_{j}(t)), t ∈ [0, 1], j = 0, 1, 2 of the three curves γ_{0}, γ_{1}, γ_{2} given by γ_{j}(t) = (t, t + (j − 1)π^{−1} sin(πt)), t ∈ [0, 1], j = 0, 1, 2. If two of these images contain the point (0, −1) on the unit circle, then that value is taken twice. If at most one of these images contains (0, −1), then by the intermediate value theorem, two images must contain (0, 1). □

As explained in [6], by results of Levitan [8], [9], Ch. 3 and Levitan-Gasymov [10], one can prove

Proposition 5.2. Suppose that x_{0} < y_{0} < x_{1} < y_{1} < ⋯ so that for n sufficiently large, x_{n} = [(2n)π]^{2}, y_{n} = [(2n + 1)π]^{2}. Then there exists a unique h_{1} and a C^{∞}-function q on [\frac{1}{2}, 1] so that

−\frac{d^{2}}{dx^{2}} + q in L^{2}((\frac{1}{2}, 1)); u′(\frac{1}{2}) = 0, u′(1) + h_{1}u(1) = 0

has eigenvalues {x_{n}}_{n=0}^{∞} and

−\frac{d^{2}}{dx^{2}} + q in L^{2}((\frac{1}{2}, 1)]; u(\frac{1}{2}) = 0, u′(1) + h_{1}u(1) = 0

has eigenvalues {y_{n}}_{n=0}^{∞}. Moreover, if a finite subset of x’s and y’s is varied, h_{1} varies continuously as a function of these numbers.

Consider now fixing y_{n} = [(2n + 1)π]^{2} for all n ∈ ℕ_{0} (= ℕ ∪ {0}) and x_{n} = [(2n)π]^{2} for n ≥ 2 and varying (x_{0}, x_{1}) in [0, 1] × [20, 21]. By Lemma 5.1 and Proposition 5.2, we can find (x_{0}^{(0)}, x_{1}^{(0)}) ≠ (x_{0}^{(1)}, x_{1}^{(1)}) so that the corresponding values of h_{1} are equal. Set q̃_{0}, q̃_{1} as the corresponding q’s and h as the common value of h_{1}. Let q_{1}, q_{2} be defined on [0, 1] by

q_{1}(x) = q̃_{0}(1 − x),  0 ≤ x ≤ \frac{1}{2},
= q̃_{1}(x),  \frac{1}{2} ≤ x ≤ 1,
q_{2}(x) = q_{0}(1 − x).

Then q_{1} ≠ q_{2} but S(0, \frac{1}{2}; h_{0} = −h, h_{\frac{1}{2}} = ∞; q_{1}) = S(0, \frac{1}{2}; h_{0} = −h, h_{\frac{1}{2}} = ∞; q_{2}) = S(\frac{1}{2}, 1; h_{\frac{1}{2}} = ∞, h_{1} = h; q_{1}) = S(\frac{1}{2}, 1; h_{\frac{1}{2}} = ∞, h_{1} = h; q_{2}) = \{((2n + 1)π)^{2}\}_{n∈ℕ}, and by reflection symmetry:

S(0, 1; h_{0} = −h, h_{1} = h; q_{1}) = S(0, 1; h_{0} = −h, h_{1} = h; q_{2}).

Since q_{1} ≠ q_{2}, this provides the required counterexample. (There is no particular significance in our choice of x_{1} ∈ [20, 21]. Any interval of length one contained in (y_{0}, y_{1}) = (π^{2}, 9π^{2}) would be admissible.)

As in the finite-difference case [5], we believe an analysis of the situation where S(0, \frac{1}{2})∩ S(\frac{1}{2}, 1) has k-points will yield k-parameter sets of q’s (as long as we are allowed to vary h_{0}, h_{1} as well as q) consistent with the given sets of eigenvalues.

§6. The Whole Line Case

In this section, we will extend Theorem 2 to the situation where [0, 1] is replaced by ℝ, but the spectrum of the corresponding Schrödinger operator H in L^{2}(ℝ) is purely discrete and bounded from below. Typical situations are, for instance, q ∈ L_{loc}^{1}(ℝ) realvalued with q(x) → ∞ as |x| → ∞ or, q ∈ L_{loc}^{1}(ℝ) real-valued, q bounded from below, and lim_{x→±∞} ∫_{x}^{x+a} dy q(y) = ∞ for any a > 0 (cf. [11], Sect. 4.1). In this case, the maximal operator H in L^{2}(ℝ) associated with the differential expression −\frac{d^{2}}{dx^{2}} + q on ℝ (with domain D(H) = {f ∈ L^{2}(ℝ) | f, f′ locally absolutely continuous on ℝ; (−f′′ + qf) ∈ L^{2}(ℝ)}) is self-adjoint. In [6] our extensions required a hypothesis on q that q(x) ≥ C|x|^{2+ε} + 1 for some C, ε > 0. This was because we used results on densities of zeros. Here, because we rely on Theorems 2.1, 2.2, we note that the following result holds by the identical proof to Theorem 2:

Theorem 3. Suppose q ∈ L_{loc}^{1}(ℝ) is real-valued and H in L^{2}(ℝ) is bounded from below with purely discrete spectrum S(−∞, ∞; q). Let S(−∞, 0; h_{0}; q) denote the spectrum of the corresponding (maximally defined) operator in L^{2}((−∞, 0)) with u′(0) + h_{0}u(0) = 0 boundary conditions, and similarly for S(0, ∞; h_{0}; q). Suppose that q_{1}, q_{2} are given and we have a fixed h_{0} ∈ ℝ ∪ {0} so that

(i) S(−∞, ∞; q_{1}) = S(−∞, ∞; q_{2}), S(−∞, 0; h_{0}; q_{1}) = S(−∞, 0; h_{0}; q_{2}), and S(0, ∞; h_{0}; q_{1}) = S(0, ∞; h_{0}; q_{2})

(ii) The sets S(−∞, ∞; q_{1}), S(−∞, 0; h_{0}; q_{1}), and S(0, ∞; h_{0}; q_{1}) are pairwise dis-

joint. Then q_{1} = q_{2} a.e. on ℝ.

As noted in Remark 2 following Theorem 2.1, this result extends to Schrödinger operators H with purely discrete spectra accumulating at +∞ and −∞. In particular, it extends to cases where H is in the limit circle case at +∞ and/or −∞ as long as the corresponding (separated) boundary condition at +∞ and/or −∞ is kept fixed for all three operators on ℝ, (−∞, 0), and (0, ∞).

The reader might want to contrast Theorem 3 with Corollary 3.4 in [3], where we obtained uniqueness of q from three (discrete) spectra of operator realizations of −\frac{d^{2}}{dx^{2}} +q on ℝ. There one of the three spectra is S(−∞, ∞; q) as above in Theorem 3; the other two, S(−∞, ∞; β_{j}, q), j = 1, 2, are associated with −\frac{d^{2}}{dx^{2}} + q on ℝ and the boundary conditions lim_{ε↓0}[u′(±ε) + β_{j}u(±ε)] = 0, where β_{j} ∈ ℝ ∪ {∞}, j = 1, 2, β_{1} ≠ β_{2}, (β_{1}, β_{2}) ≠ (0, ∞), (∞, 0).

References

1. G. Borg, Uniqueness theorems in the spectral theory of y′′ +(λ−q(x))y = 0, Proc. 11th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag, Oslo, 1952, 276–287.

2. R. del Rio, F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Intl. Math. Research Notices, to appear.

3. F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for onedimensional Schrödinger operators, Trans. Amer. Math. Soc. 348 (1996), 349–373.

4. F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, I. The case of an a.c. component in the spectrum, Helv. Phys. Acta 70 (1997), 66–71.

5. F. Gesztesy and B. Simon, m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, to appear in J. d’Anal. Math.

6. F. Gesztesy and B. Simon Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, preprint, 1997.

7. B. Ja. Levin, Distribution of Zeros of Entire Functions, rev. ed., Amer. Math. Soc., Providence, RI, 1980.

8. B. M. Levitan, On the determination of a Sturm-Liouville equation by two spectra, Amer. Math. Soc. Transl. 68 (1968), 1–20.

9. B. M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.

10. B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two of its spectra, Russ. Math. Surv. 19:2 (1964), 1–63.

11. B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory, Amer. Math. Soc., Providence, RI, 1975.

12. V. A. Marchenko, Some questions in the theory of one-dimensional linear differential operators of the second order, I, Trudy Moskov. Mat. Obšč. 1 (1952), 327–420 (Russian); English transl. in Amer. Math. Soc. Transl. (2) 101 (1973), 1–104.

13. V. N. Pivovarchik, An inverse Sturm-Liouville problem by three spectra, unpublished.

14. B. Simon, A new approach to inverse spectral theory, I. Fundamental formalism, in preparation.

15. E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press, Oxford,