INVERSE SPECTRAL ANALYSIS WITH PARTIAL INFORMATION

ON THE POTENTIAL, III. UPDATING BOUNDARY CONDITIONS

Rafael del Rio^{1}, Fritz Gesztesy^{2}, and Barry Simon^{3}

June 3, 1997

Abstract. We discuss results where information on parts of the discrete spectra of onedimensional Schrödinger operators H = − \frac{d^{2}}{dx^{2}} + q in L^{2}((0, 1)) or of a finite Jacobi matrix together with partial information on q uniquely determines q a.e. on [0, 1]. These extend classical results of Borg and Hochstadt-Lieberman as well as results in paper II of this series.

§1. Introduction

This paper is a postscript to two earlier papers [5, 6] in that it provides a new way of looking at the problems considered in those papers that allows the same methods to prove additional results.

To explain our results, we recall earlier theorems of Borg [1] (see also [8, 10–14]) and of Hochstadt-Lieberman [9] (see also [7, 15]). Throughout this paper assume q ∈ L^{1}((0, 1)) to be real-valued and consider the operator H = −\frac{d^{2}}{dx^{2}} + q in L^{2}((0, 1)) with boundary conditions

u′(0) + h_{0}u(0) = 0,  (1.1)
u′(1) + h_{1}u(1) = 0,  (1.2)

where h_{j} ∈ ℝ ∪ {∞}, j = 0, 1 (with h_{0} = ∞ shorthand for the boundary condition u(0) = 0). Fix h_{1} ∈ ℝ but think of H(h_{0}) as a family of operators depending on h_{0} as a parameter. Then Borg’s and Hochstadt-Lieberman’s results can be paraphrased as follows:

1 IIMAS-UNAM, Apdo. Postal 20-726, Admon No. 20, 01000 Mexico D.F., Mexico. E-mail: delrio@servidor.unam.mx

2 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: fritz@math.missouri.edu 3 Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasade-na,

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

This material is based upon work supported by CONACYT Project 05567P-E and the National Science Foundation under Grant Nos. DMS-9623121 and DMS-9401491.

To be submitted to Intl. Math. Research Notes

Typeset by AMS-TEX

Borg [1]. The spectra of H(h_{0}) for two values of h_{0} determine q.

Hochstadt-Lieberman [9]. The spectra of H(h_{0}) for one value of h_{0} and q on [0, \frac{1}{2}] determine q.

In [6], two of us proved a result that can be paraphrased as

Theorem of [6]. Half the spectra of one H(h_{0}) and q on [0, \frac{3}{4}] determine q.

One of our goals in this note is to prove

New Result. The spectrum of one H(h_{0}) and half the spectrum of another H(h_{0}) and q on [0, \frac{1}{4}] determine q.

We will also show that

New Result. Two-thirds of the spectra of three H(h_{0}) determine q.

Our point is as much a new way of looking at the argument in [6] as these new results. Fundamental to our approach here and in [5, 6] is the Titchmarsh-Weyl m-function defined by

m_{h_{1}}(z) = \frac{u′_{h_{1}}(z, 0)}{u_{h_{1}}(z, 0)} ,

where u_{h_{1}}(z, x) solves −u′′(z, x) + q(x)u(z, x) = zu(z, x) with the boundary condition (1.2). m_{h_{1}} is a meromorphic function on ℂ (in fact, a Herglotz function) with all its zeros and poles on the real axis. Since h_{1} ∈ ℝ will be fixed throughout this paper, we will delete the subscript h_{1} from now on and simply write m(z) instead. Moreover, due to the assumption h_{1} ∈ ℝ, we will index the eigenvalues of H(h_{0}) by {λ_{n}}_{n∈ℕ_{0}}, ℕ_{0} = ℕ ∪ {0}.

A fundamental result of Marchenko [16] (see also [2, 3, 17]) says

Theorem 1.1. m(z) uniquely determines q a.e. on [0, 1].

Our fundamental strategy can be described as follows:

(a) Note that λ is an eigenvalue of H(h_{0}) if and only if m(λ) = −h_{0}.

(b) Prove a general theorem that knowing m at points λ_{0}, λ_{1}, …, λ_{n}, … determines m as long as {λ_{n}}_{n∈ℕ_{0}} has sufficient density. Given (a), this will allow one to prove that if λ_{0}, λ_{1}, …, λ_{n}, … have sufficient density, an infinite sequence of pairs {(λ_{n}, α_{n})}_{n∈ℕ_{0}} and the knowledge that H(h_{0} = α_{n}) has an eigenvalue at λ_{n} determines m (and so q a.e. on [0, 1] by Theorem 1.1).

(c) Use scaling covariance to extend the [0, 1] result to one for [x, 1] for any x ∈ (0, 1).

(d) Note that a knowledge of q a.e. on [0, x] allows one to update boundary conditions. Explicitly, let H(h_{x}) be the operator in L^{2}((x, 1)) with boundary condition (1.2) but (1.1) replaced by

u′(x) + h_{x}u(x) = 0.  (1.3)

Then λ_{n} is an eigenvalue of H(h_{0} = α_{n}) if and only if it is an eigenvalue of H(h_{x_{0}} = β_{n}), where β_{n} is obtained by solving m′_{n}(x) = q(x) − λ_{n} − m_{n}^{2} on [0, x_{0}] with the boundary condition m_{n}(x = 0) = −α_{n} and setting β_{n} = −m_{n}(x = x_{0}).

We will present steps (b) and (c) in Sections 2 and 3 and then step (d) in Section 4. We will not explicitly derive them, but the results in [6] that treat operators on (0, 1)

and that allow one to trade C^{2k} conditions on q for k eigenvalues can be extended to the context we discuss here.

We also note that the ideas in this paper extend to Jacobi matrices.

Finally, while the present paper and [5, 6] concentrate on discrete spectra, we might point out that our m-function strategy also applies in certain cases involving absolutely continuous spectra, see [4].

§2. Zeros of the m-function

If a ∈ ℝ, let a_{+} = max(a, 0). Then

Theorem 2.1. Let {λ_{n}}_{n∈ℕ_{0}} be a sequence of distinct positive real numbers satisfying

∑_{n=0}^{∞} \frac{(λ_{n} − \frac{1}{4}π^{2}n^{2})_{+}}{n^{2}} < ∞.  (2.1)

Let m_{1}, m_{2} be the m-functions for two operators H_{j} = −\frac{d^{2}}{dx^{2}} + q_{j} in L^{2}((0, 1)) with boundary conditions

u′(1) + h_{1}^{(j)}u(1) = 0

and h_{1}^{(j)} ∈ ℝ, j = 1, 2. Suppose that m_{1}(λ_{n}) = m_{2}(λ_{n}) for all n ∈ ℕ_{0}. Then m_{1} = m_{2} (and hence q_{1} = q_{2} a.e. on [0, 1] and h_{1}^{(1)} = h_{1}^{(2)}).

Remarks. 1. In our examples, λ_{n} ∼ π^{2}n^{2} + C as n → ∞ (cf. (3.1)), so (2.1) is satisfied, for instance, by considering two distinct spectra of H(h_{0}).

2. We allow the case m_{1}(λ_{n}) = m_{2}(λ_{n}) = ∞.

As a preliminary result we note the following

Lemma 2.2. Suppose {λ_{n}}_{n∈ℕ_{0}} is a sequence of positive real numbers satisfying (2.1) and

∑_{n=0}^{∞} λ_{n}^{−1} < ∞.  (2.2)

Define f(z) := ∏_{n=0}^{∞}(1 − \frac{z}{λ_{n}}), then

\overline{lim}_{\begin{matrix}|y|→∞ \\ y∈ℝ \end{matrix}} \frac{|y|^{1/2} sinh(2|y|^{1/2})}{|f(iy)|} < ∞.  (2.3)

Proof. Let y ∈ ℝ. Then sinh(2|y|^{1/2} )/|y|^{1/2} = | sin(2i|y|^{1/2})/|y|^{1/2}| and

\frac{sin(2\sqrt{z})}{2\sqrt{z}} = ∏_{n=1}^{∞}(1 − \frac{4z}{π^{2}n^{2}} ),

so (2.3) becomes

\overline{lim}_{|y|→∞} \frac{|y|}{1 + \frac{|y|}{λ_{0}}} ∏_{n=1}^{∞}[\frac{(1 + \frac{4|y|}{π^{2}n^{2}} )}{(1 + \frac{|y|}{λ_{n}})}] < ∞  (2.4)

using 2^{−1/2}(1 + |x|) ≤ (1 + x^{2})^{1/2} ≤ (1 + |x|).

If 0 ≤ a ≤ b, then (\frac{1+a|y|}{1+b|y|}) ≤ 1, and if a > b > 0, then

\frac{(1 + a|y|)}{1 + b|y|} = 1 + \frac{(a − b)|y|}{1 + b|y|} ≤ 1 + \frac{a − b}{b} = \frac{a}{b},

∏_{n=1}^{∞} \frac{(1 + \frac{4|y|}{π^{2}n^{2}} )}{(1 + \frac{|y|}{λ_{n}})} ≤ ∏_{n:λ_{n}> \frac{1}{4} π^{2}n^{2}} \frac{4λ_{n}}{π^{2}n^{2}} = ∏_{n=1}^{∞}[1 + \frac{(λ_{n} − \frac{1}{4}π^{2}n^{2})_{+}}{\frac{1}{4}π^{2}n^{2}} ] < ∞

if (2.1) holds. □

Proof of Theorem 2.1. We follow the arguments in [5, 6] fairly closely. One can write m_{j}(z) = \frac{Q_{j}(z)}{P_{j}(z)}, j = 1, 2, where

(1) P_{j}, Q_{j} are entire functions satisfying

|P_{j}(z)| ≤ C exp(\sqrt{|z|}),  (2.5a)
|Q_{j}(z)| ≤ C(1 + \sqrt{|z|}) exp(\sqrt{|z|}).  (2.5b)

(2)

m_{j}(z) = ±i\sqrt{z} + o(1) as z → ±i∞.  (2.6)

(We use the square root branch with Im (\sqrt{z}) ≥ 0.)

Suppose m_{1} ≠ m_{2}. Then P_{2}(z)Q_{1}(z)−P_{1}(z)Q_{2}(z) := H(z) is an entire function of order at most \frac{1}{2} and not identically zero. Since H(λ_{n}) = 0, we conclude that ∑_{n∈ℕ_{0}} λ_{n}^{−a} < ∞ if a > \frac{1}{2}. In particular, (2.2) holds, and we can define f(z) = ∏_{n=0}^{∞}(1 − \frac{z}{λ_{n}}). Next, define

G(z) := \frac{H(z)}{f(z)} = \frac{P_{1}(z)P_{2}(z)}{f(z)} (m_{1}(z) − m_{2}(z)).  (2.7)

Since H(λ_{n}) = 0, G(z) is an entire function. By (2.3),

\overline{lim}_{|y|→∞} \frac{|y|^{1/2} exp (2|y|^{1/2})}{|f(iy)|} < ∞,

so by (2.5) and (2.6),

|G(iy)| ≤ \frac{exp (2|y|^{1/2})}{f(iy)} |m_{1}(iy) − m_{2}(iy)| = o(|y|^{−1/2})

goes to zero as |y| → ∞. The Phragmén-Lindelöf argument of [6] then yields the contradiction G(z) ≡ 0, that is, m_{1} = m_{2}. □

Remark. The above yields o(|y|^{−1/2}) even though o(1) would have been sufficient. We have thrown away half a zero. That means one can prove the following result.

Theorem 2.2. Let {λ_{n}}_{n∈ℕ_{0}} and {μ_{n}}_{n∈ℕ_{0}} be two sequences of real numbers satisfying

∑_{n=0}^{∞} \frac{(λ_{n} − π^{2}n^{2})_{+}}{n^{2}} < ∞  and  ∑_{n=0}^{∞} \frac{(μ_{n} − π^{2}n^{2})_{+}}{n^{2}} < ∞,  (2.8)

with μ_{m} ≠ λ_{n} for all m, n ∈ ℕ_{0}. Let m_{1}, m_{2} be the m-functions for two operators H_{j} = −\frac{d^{2}}{dx^{2}} + q_{j}, j = 1, 2 in L^{2}((0, 1)) with boundary conditions

u′(1) + h_{1}^{(j)}u(1) = 0

and h_{1}^{(j)} ∈ ℝ, j = 1, 2. Suppose that m_{1}(z) = m_{2}(z) for all z in {λ_{n}}_{n=0}^{∞} ∪ {μ_{n}}_{n=0}^{∞} except perhaps for one. Then m_{1} = m_{2} (and hence q_{1} = q_{2} a.e. on [0, 1] and h_{1}^{(1)} = h_{1}^{(2)}).

By scaling, one sees the following analog of Theorem 2.1 holds (there is also an analog of Theorem 2.2):

Theorem 2.3. Let a < b and {λ_{n}}_{n∈ℕ_{0}} be a sequence of distinct positive real numbers satisfying

∑_{n=0}^{∞} \frac{(λ_{n} − \frac{π^{2}n^{2}}{4(b−a)^{2}} )_{+}}{n^{2}} < ∞.  (2.9)

Let m_{1}, m_{2} be the m-functions for two operators H_{j} = −\frac{d^{2}}{dx^{2}} + q_{j}, j = 1, 2 in L^{2}((a, b)) with boundary conditions (1.3) at x = a and

u′(b) + h_{b}^{(j)}u(b) = 0,

where h_{b}^{(j)} ∈ ℝ, j = 1, 2. Suppose that m_{1}(λ_{n}) = m_{2}(λ_{n}) for all n ∈ ℕ_{0}. Then m_{1} = m_{2} (and hence q_{1} = q_{2} a.e. on [a, b] and h_{b}^{(1)} = h_{b}^{(2)}).

§3. Whole Interval Results

Fix h_{1} ∈ ℝ, let H(h_{0}) be the operator on L^{2}((0, 1)) with u′(1)+h_{1}u(1) = 0 and u′(0)+ h_{0}u(0) = 0 boundary conditions, and denote by λ_{n}(h_{0}) the corresponding eigenvalues of H(h_{0}). Then, for h_{0} ∈ ℝ, it is known (see, e.g., the references in [6]) that

λ_{n} = (nπ)^{2} + 2(h_{1} − h_{0}) + ∫_{0}^{1} q(x) dx + o(1) as n → ∞  (3.1)

and for h_{0} = ∞,

λ_{n} = [(n + \frac{1}{2})π]^{2} + 2h_{1} + ∫_{0}^{1} q(x) dx + o(1) as n → ∞.  (3.2)

To say that H(h_{0}) has eigenvalue λ is equivalent to m(λ) = −h_{0}. Thus, Theorem 2.1 implies Theorem 3.1. Let H_{1}(h_{0}), H_{2}(h_{0}) be associated with two potentials q_{1}, q_{2} on [0, 1] and two potentially distinct boundary conditions h_{1}^{(1)}, h_{1}^{(2)} ∈ ℝ at x = 1. Suppose that {(λ_{n}, h_{0}^{(n)})}_{n∈ℕ_{0}} is a sequence of pairs with λ_{0} < λ_{1} < ⋯ → ∞ and h_{0}^{(n)} ∈ ℝ ∪ {∞} so that both H_{1}(h_{0}^{(n)}) and H_{2}(h_{0}^{(n)}) have eigenvalues at λ_{n}. Suppose that (2.1) holds. Then q_{1} = q_{2} a.e. on [0, 1] and h_{1}^{(1)} = h_{1}^{(2)}.

Given (3.1), (3.2) we immediately have Borg’s theorem [1] as a corollary (this is essentially the usual proof), but more is true. For example, by using Theorem 2.2 one infers:

Corollary 3.2 [1]. Fix h_{0}^{(1)}, h_{0}^{(2)} ∈ ℝ. Then all the eigenvalues of H(h_{0}^{(1)}) and all the eigenvalues of H(h_{0}^{(2)}), save one, uniquely determine q a.e. on [0, 1].

Corollary 3.3. Let h_{0}^{(1)}, h_{0}^{(2)}, h_{0}^{(3)} ∈ ℝ and denote by σ_{j} = σ(H(h_{0}^{(j)})) the spectra of H(h_{0}^{(j)}), j = 1, 2, 3. Assume S_{j} ⊆ σ_{j}, j = 1, 2, 3 and suppose that for all sufficiently large λ_{0} > 0 we have

#\{λ ∈ \{S_{1} ∪ S_{2} ∪ S_{3}\} with λ ≤ λ_{0}\} ≥ \frac{2}{3}#\{λ ∈ \{σ_{1} ∪ σ_{2} ∪ σ_{3}\} with λ ≤ λ_{0}\} − 1.

Then q is uniquely determined a.e. on [0, 1].

In particular, two-thirds of three spectra determine q.

§4. Updating m

We are now able to understand why partial information on q — knowing it on [0, a] — lets us get away with less information on eigenvalues, a phenomenon originally discovered by Hochstadt-Lieberman [9] in the special case where a = \frac{1}{2}. We note that m(z, x) satisfies the Ricatti-type equation

m′(z, x) = q(x) − z − m^{2}(z, x).  (4.1)

If we know that λ is an eigenvalue of H(h_{0}), then m(λ, 0) = −h_{0}. If we know q on [0, a], we can use (4.1) to compute m(λ, a) := −h_{a} and so infer that λ is an eigenvalue of H(h_{a}), the operator in L^{2}((a, 1)). By Theorem 2.3, that means we only need a lower density of eigenvalues of the various H(h_{a}). A typical result is the following

Theorem 4.1. Let σ_{N} and σ_{D} be the eigenvalues of H(h_{0} = 0) and H(h_{0} = ∞), respectively. Let S_{N} ⊆ σ_{N}, S_{D} ⊆ σ_{D}. Fix a ∈ (0, 1). Suppose for λ_{0} > 0 sufficiently large that

#{λ ∈ {S_{N} ∪ S_{D}} with λ ≤ λ_{0}} ≥ (1 − a)#{λ ∈ {σ_{N} ∪ σ_{D}} with λ ≤ λ_{0}}.

Then S_{N}, S_{D} and q on [0, a] uniquely determine q a.e. on [0, 1].

This follows immediately from the updating idea. For example, if a = \frac{3}{4}, we can recover Theorem 1.3 of [6] (it is essentially a reworking of the proof in [6]); but for a ∈ (0, \frac{1}{2}), the result is new and implies, for example, that q on [0, \frac{1}{4}], all the Neumann eigenvalues, and half the Dirichlet eigenvalues determine q a.e. on [0, 1].

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