Research paperA nodal inverse problem for measure geometric Laplacians
Introduction
Let μ be an atomless probability Borel measure supported on a compact set Kμ ⊂ [0, 1], and let us call the associated measure geometric Laplacian, introduced by Feller and Krein [8], [18] in the 50s, and widely studied in the last years by Bird, Deng, Freiberg, Löbus, Ngai, Teplayev, among many others authors, see [3], [4], [6], [9], [10], [11], [24].
We are interested here in the information contained in the nodal sets of the eigenfunctions of the eigenvalue problem
In Section 2 we introduce the notation we use and relevant definitions, together with a short review of the main results on the eigenvalue problem (1.1) that will be needed later.
Then, in Section 3 we consider the following inverse nodal problems:
Problem 1, Uniqueness: Let be the set of nodal points of the eigenfunctions {φn}n ≥ 1 of problem (1.1). Then, if show that .
Problem 2, Recovery: Given the set of nodal points of the eigenfunctions {φn}n ≥ 1 of problem (1.1), characterizeμ.
Both problems originated in the theory of orthogonal polynomials, see the classical books of Szego [26] and Freud [13]. However, for classical ordinary differential equations this problems were introduced by Hald and McLaughlin in [15], [16], [21]. Observe that this correspond to the Lebesgue measure and the absolutely continuous measures with ρ ∈ BV([0, 1]). Initially, a two term asymptotic expansion of eigenvalues was needed, together with a positive lower bound of ρ, and higher regularity of ρ, namely . From this asymptotic, it is possible to derive a precise estimate of the distance between two consecutive zeros, and since this is known from the nodal data, we can reverse the process in order to estimate the weight ρ. Notice that the distance between zeros of φn goes to zero when n → ∞ for ordinary differential equations. Later, in [22], [23] we showed that was enough to characterize ρ and we were able to remove the condition ρ > c > 0, extending the result to sign changing weights and unbounded intervals. Let us observe that in these problems the distance between two consecutive zeros does not go to zero as before, as well as for measure geometric Laplacians.
To our knowledge, the only results about zeros of eigenfunctions of geometric measure Laplacians, for infinite Bernoulli convolutions, are due to Freiberg and Löbus in [10]. They showed that the asymptotic distribution of zeros is a Cantor-type staircase function, which is the cumulated distribution of another infinite Bernoulli convolution with the same support and different parameters. So, given Sμ, their result implies that you can identify μ in this class of measures.
In Section 3 we start the analysis of Problems 1 and 2 for general measures. Our first result is the following theorem, which states that the zeros of the eigenfunctions concentrate on the support of the measure Kμ. Theorem 1.1 Let φn be an eigenfunction of the problem (1.1), and let be their nodal points. Let us introduce the discrete probability measuresand let us call the cumulate distribution function,Then there exists a subsequence of {Zn}n ≥ 1 which converges point-wise to . Moreover, the function Z is constant in the complement of Kμ.
Observe that in [10], . However, this is not always the case, and after the proof we show an example where Z is constant on a substantial part of Kμ, see Example 3.1. Hence, a natural problem is to recover the missing points Kμ∖KdZ, and we need to perform a local study of the problem.
Let us defineand we have the following result: Theorem 1.2 Let be the set of nodal points of the eigenfunctions {φn}n ≥ 1 of problem (1.1). Let and . Then, for any δ < η, Nn(x0, δ) ≤ 1. Let x0 ∈ Kμ. Then, for any δ > 0,
Therefore, the local analysis enable us to identify Kμ. However, different measures can have the same support.
In the following results we fix any measure μ, and we consider the class of absolutely continuous measures with respect to μ with a continuous Radon-Nikodym derivative. Our next theorems settles Problem 1 and 2 for this class of measures. Theorem 1.3 Let where ν ≪ μ are probability measures with Radon-Nikodym derivative ρ ∈ C(Kμ). Then, ρ ≡ 1. Theorem 1.4 Given μ and Sν, where ν ≪ μ are probability measures with Radon-Nikodym derivative ρ ∈ C(Kμ), we recover ρ from the nodal set Sν. Remark 1.5 We do not use the asymptotic behavior of the eigenvalues in the proof, and the arguments we use to find ρ are new even in the classical setting, since we are not assuming the existence of an asymptotic formula for the growth of the eigenvalues. Our proof was motivated by the dichotomy between the arithmetic/non arithmetic case (see [17]), since for the non-arithmetic case the best we can expect is an estimate where ds is the so-called spectral dimension, which is not enough to prove uniqueness nor to characterize the weight using previous ideas in [12], [15], [16], [19], [20], [21], [22], [23], [25]. Remark 1.6 Let us remark also that the restriction to probability measures can be changed by considering another kind of normalization, or knowing one of the eigenvalues. Briefly, given a positive constant c, the eigenvalues of the measure geometric Laplacians Δμ and Δcμ are scaled by a factor c, and have exactly the same eigenfunctions, so the nodal sets coincide.
Finally, we conclude in Section 4 by describing possible extensions and open problems.
Section snippets
Notations and measures
In this work we deal with atomless Borel probability measures, denoted by μ and ν. We assume that μ has bounded support Kμ, and that Kμ ⊂ [0, 1]. We denote by its complement, [0, 1]∖Kμ.
Let us recall that ν is absolutely continuous with respect to μ in [0,1], denoted by ν ≪ μ, if there exists some function ρ ∈ L1([0, 1], μ), the Radon-Nikodym derivative, such that, for any [c, d] ⊂ [0, 1], we have
We denote by δx the Dirac’s delta measure, satisfying if x ∈ A, and
Concentration of zeros
Let us prove now the concentration of zeros on Kμ. Proof of Theorem 1.1 Since all the measures dZn are concentrated in [0,1], the family of probability measures {dZn}n ≥ 1 is tight, and we can extract a subsequence of measures that converge to some probability measure dZ. Hence, by Helly’s selection theorem, there exists a subsequence of the cumulate distribution functions that converges point-wise to a real function Z. Now, let us consider . Since any eigenfunction φn is lineal in (a, b), it has at
Final remarks
We have considered the nodal inverse problem for measure geometric Laplacians. The proof of Theorem 1.4 extend the results obtained in the classical setting, and provides a new approach which can be used in other problems, specially when the asymptotic behavior of the eigenvalues is not known.
Few simple generalizations are possible, and we can expect that Theorem 1.4 holds for Bounded Variations weights ρ, other boundary conditions, and even to discover the kind of boundary condition
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was partially supported by Universidad de Buenos Aires 20020170100445BA, by Agencia Nacional de Promoción Científica y Técnica PICT 2016-1022, and CONICET PIP 11220150100032CO.
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Both authors have contributed equally.