CONNECTEDNESS OF THE ISOSPECTRAL MANIFOLD FOR ONE-DIMENSIONAL HALF-LINE

SCHRÖDINGER OPERATORS

FRITZ GESZTESY^{1} AND BARRY SIMON^{2}

To Elliott Lieb on his 70th birthday, with our best wishes

Abstract. Let V_{0} be a real-valued function on [0, ∞) and V ∈ L^{1}([0, R]) for all R > 0 so that H(V_{0}) = −\frac{d^{2}}{dx^{2}} + V_{0} in L^{2}([0, ∞)) with u(0) = 0 boundary conditions has discrete spectrum bounded from below. Let M(V_{0}) be the set of V so that H(V) and H(V_{0}) have the same spectrum. We prove that M(V_{0}) is connected.

The limitation of our knowledge of inverse spectral theory for Schrödinger operators H_{ℝ}(V) = −\frac{d^{2}}{dx^{2}} + V in L^{2}(ℝ) is shown by the following open question: Let M_{ℝ}(x^{2}) denote the set of all V ’s with σ(H_{ℝ}(V)) = {1, 3, 5, … }, the spectrum of the harmonic oscillator −\frac{d^{2}}{dx^{2}} + x^{2} in L^{2}(ℝ). Is M_{ℝ}(x^{2}) connected? One can ask the same question for restricted sets of V ’s, say requiring that V is C^{k}(ℝ) for some k ∈ ℕ or C^{∞}(ℝ). The question remains open for sets so large that they include the translates of V.

If one demands that V be close to x^{2} in a strong sense, there are some results that go back to McKean–Trubowitz [9] (see also Levitan [7]), culminating in the recent beautiful paper of Chelkak, Kargaev, and Korotyaev [1] who require V (x) = x^{2} + q(x) with ∫ [|q′(x)|^{2} + x^{2}|q^{2}(x)|^{2}] dx < ∞ (by a Sobolev estimate such q’s have |q(x)| → 0 as |x| → ∞). Their analysis implies the set of V ∈ M_{ℝ}(x^{2}) obeying this estimate is connected.

The purpose of this note is to make what turns out to be an elementary observation: The corresponding problem for the half-line is easy! Suppose V is real-valued and in L^{1}([0, R]) for all R > 0. We consider H(V) = −\frac{d^{2}}{dx^{2}} + V in L^{2}([0, ∞)) with u(0) = 0 (i.e., Dirichlet) boundary conditions. If V is merely in L^{1}([0, R]) for all R > 0, we will use that to define a topology on M(V). If V is in C^{k}([0, ∞)) for some k ∈ ℕ, we will use the norm on C^{k}([0, R]) for all R > 0 to define the topology on M(V). Here is our result:

Date: June 19, 2003.

2000 Mathematics Subject Classification. Primary: 34A55, 34L40, Secondary:

34B20, 34B24.

Key words and phrases. Isospectral sets of potentials, half-line Schrödinger operators, inverse problems.

1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: fritz@math.missouri.edu.

2 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

Theorem 1. Let V_{0}, V_{1} be real-valued and in L^{1}([0, R]) for all R > 0 so that H(V_{ℓ}) is bounded from below (which means that −\frac{d^{2}}{dx^{2}} + V_{ℓ} is in the limit point case at ∞, cf. [5]) and each has discrete spectrum spec(H(V_{ℓ})) = {E_{j}(V_{ℓ})}_{j∈ℕ}, ℓ = 0, 1.

Suppose

spec(H(V_{0})) = spec(H(V_{1})).  (1)

Then there exists {V_{t}}_{0≤t≤1}, with V_{t} real-valued and in L^{1}([0, R]) for all R > 0, interpolating V_{0} and V_{1} so that

spec(H(V_{t})) = spec(H(V_{0})), t ∈ [0, 1]

and t → V_{t}|_{[0,R]} is continuous in L^{1}([0, R]) for all R > 0. Moreover,

(i) t → V_{t}|_{[0,R]} is real analytic in L^{1}([0, R]) for all R > 0. (ii) If V_{0} and V_{1} are C^{k}([0, ∞)) for some k ∈ ℕ, then V_{t} is C^{k}([0, ∞))

and t → V_{t}|_{[0,R]} is real analytic in C^{k}([0, R])) for all R > 0.

We recall that all eigenvalues {E_{j}(V)}_{j∈ℕ} of the Dirichlet operator H(V) in L^{2}([0, ∞)) are simple.

The proof exploits the A-function studied by us in [4, 13]. A can be defined in terms of the spectral measure defined in the standard way (see, e.g., [8]) so that the Weyl m-function satisfies

m(z) = c + ∫_{ℝ} [\frac{1}{λ − z} − \frac{λ}{1 + λ^{2}} ] dρ(λ), z ∈ ℂ\spec(H(V)). (2)

We let dρ_{0} denote the spectral measure for the case V = 0. It is well-known, since

m(z, V = 0) = i\sqrt{z},  (3)

that

dρ_{0}(E) = \frac{1}{π} \sqrt{E} χ_{[0,∞)}(E) dE.  (4)

A is then defined by

A(α) = −2 ∫_{−∞}^{∞} λ^{− \frac{1}{2}} sin(2α\sqrt{λ}) [dρ(λ) − dρ_{0}(λ)],  (5)

where the integral is intended in the distributional sense on (−∞, ∞) (so, a priori, A is only a distribution, not a function). Of course, A = 0 for V = 0.

One can also define a distribution A by (5) with ρ_{0} dropped (cf. [4]) but then this distribution is only L_{loc}^{1} away from 0 and has a δ′ singularity at 0. We will not use this approach in this note.

The fact on which Theorem 1 depends is the following:

Theorem 2. dρ is the spectral measure of an H(V) with V ∈ L^{1}([0, R]) for all R > 0, V real-valued, if and only if A ∈ L_{loc}^{1}(ℝ). V ∈ C^{k}([0, ∞)) for some k ∈ ℕ if and only if A ∈ C^{k}(ℝ). If dρ_{t} is a family so that A_{t}|_{[−R,R]} is real analytic in t in L^{1}([−R, R]) (resp. C^{k}([−R, R])) for all R > 0, then t → V_{t} is real analytic in t in L^{1}([0, R]) (resp. C^{k}([0, R])) for all R > 0.

This theorem combines results from Gesztesy–Simon [4] and Simon [13] (who show V is C^{k}([0, ∞)) if and only if A is, once one knows V exists) and work of Remling [10] or a suitable version of the Gel’fand– Levitan theory [6, Ch. 2], [8, Sect. 2.3] to get the existence part of Theorem 2.

Once one has Theorem 2, Theorem 1 is immediate.

Proof of Theorem 1. Let {E_{j}}_{j∈ℕ} be the common spectrum of V_{0} and V_{1} so

dρ_{ℓ}(E) = ∑_{j∈ℕ}a_{j,ℓ} δ(E − E_{j}), ℓ = 1, 2.  (6)

Define

dρ_{t}(E) = ∑_{j∈ℕ}[ta_{j,1} + (1 − t)a_{j,0}]δ(E − E_{j}), t ∈ [0, 1].

The associated A-functions satisfy

A_{t} = tA_{1} + (1 − t)A_{0}, t ∈ [0, 1].  (7)

Clearly, if A_{0}, A_{1} are L_{loc}^{1}(ℝ), so is A_{t}, and if A_{0}, A_{1} are C^{k}(ℝ), so are A_{t}. Thus, dρ_{t} is the spectral measure of a potential V_{t} that, by Theorem 2, has the claimed properties. □

Remarks. 1. The C^{k} result extends to C^{∞} and one can also extend it to real analyticity.

2. Because V on [0, x_{0}] only depends on A on [0, x_{0}] (see [4, 13]), if V_{0} and V_{1} are C^{k} on [0, x_{0}], so is each V_{t}.

The key to our proof is the fact that, while not all measures ∑_{j∈ℕ} a_{j} δ(E −E_{j}) are spectral measures (the fact that the A-transform of ρ − ρ_{0} has no singularity at α = 0, which means the ρ-term alone has a specific singularity that places restrictions on the a_{j}), those that are form a convex, hence, connected set.

The difficulty of extending this to potentials on ℝ is that there is no known way to describe when a candidate spectral measure is, in fact, the spectral measure for a potential on ℝ. Typically, one reduces the whole-line problem on ℝ to two half-line problems on (−∞, x_{0}) and (x_{0}, ∞) for some fixed x_{0} ∈ ℝ (cf. [6, Ch. 7], [11, 12]) but, in general, loses control over the potential at the point x_{0} (in the sense that generally the potential V will be discontinuous at x = x_{0}). To determine the potential, the spectral measure is a 2 × 2 matrix in this case [2, Sect. 9.5], [3, Sect. X.III.5], and because of restrictions on this matrix, convex combinations of the matrix measures coming from a potential will not come from a potential, so our method cannot extend.

We, of course, believe that for the whole-line case, there is also a result on connectedness of the spectral manifold for a potential like V (x) = x^{2}. But we wonder about a case like

V (x) = ln(|x| + 1) + exp(x)  (8)

with very different asymptotics if x → ∞ and x → −∞. V (x) and V (−x) are obviously isospectral, but we wonder if there is a path between them in the isospectral manifold. It might be that the correct conjecture is that for any V, M_{ℝ}(V) consists of either one or two connected components.

We hope that this note will stimulate more work on this problem. We close by thanking Elliott Lieb for his many years of research

results, insights, and service to the mathematics and physics communities, and by expressing the hope that he has enjoyed this birthday bouquet.

References

[1] D. Chelkak, P. Kargaev, and E. Korotyaev, The inverse problem for the harmonic oscillator perturbed by a potential. Characterization, preprint, 2003.

[2] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, Malabar, FL, 1985.

[3] N. Dunford and J.T. Schwartz, Linear Operators Part II: Spectral Theory, Interscience, New York, 1988.

[4] F. Gesztesy and B. Simon, A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure, Ann. of Math. (2) 152 (2000), no. 2, 593–643.

[5] P. Hartman Differential equations with non-oscillatory eigenfunctions, Duke Math. J. 15 (1948), 697–709.

[6] B.M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.

[7] B. Levitan, Sturm-Liouville operators on the entire real axis with the same discrete spectrum, Math. USSR-Sb. 60 (1988), no. 1, 77–106.

[8] V.A. Marchenko, Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications, 22, Birkhäuser, Basel, 1986.

[9] H.P. McKean and E. Trubowitz, The spectral class of the quantum-mechanical harmonic oscillator, Comm. Math. Phys. 82 (1981/82), no. 4, 471–495.

[10] C. Remling, Schrödinger operators and de Branges spaces, J. Funct. Anal. 196 (2002), no. 2, 323–394.

[11] F.S. Rofe-Beketov, The spectral matrix and the inverse Sturm-Liouville problem on the axis (−∞, ∞), Teor. Funktsiĭ Funkts. Analiz Prilozh. 4 (1967), 189–197. [Russian]

[12] F.S. Rofe-Beketov, The inverse Sturm-Liouville problem for the spectral matrix on the whole axis and associated problems, in Integral Equations and Inverse Problems, (V. Petkov and R. Lazarov, eds.), pp. 234–238, Longman, New York, 1991.

[13] B. Simon, A new approach to inverse spectral theory. I. Fundamental formalism, Ann. of Math. (2) 150 (1999), no. 3, 1029–1057.