Inverse spectral problem for the Euler–Bernoulli matrix beam with general measures: recovering the two mass measures from the Weyl function

The Euler–Bernoulli beam equation on a finite interval, rewritten as an inhomogeneous matrix string problem $D^2\varphi = \lambda M\varphi$ with off-diagonal measure-valued density $M = \begin{pmatrix} 0 & n \\ m & 0 \end{pmatrix}$, was introduced by Beals and Szmigielski together with a two-component Camassa–Holm-type system of which it is the isospectral spectral problem. They solved the corresponding inverse spectral problem — recovery of the two measures from the Dirichlet Weyl function $W(\lambda) = \frac{1}{\lambda} D\Phi(1,\lambda)\Phi(1,\lambda)^{-1}$ — only for discrete beams with finitely many point masses, using a non-commutative generalization of Stieltjes' continued fractions and Hankel-type determinant formulas. The open general case is to characterize the matrix functions $W$ arising from arbitrary pairs of positive Borel measures $(m,n)$, decide whether $W$ determines $(m,n)$ uniquely, and construct a recovery procedure for general measures, together with the supporting spectral theory (the infinite-spectrum partial-fraction expansion of $W$ and convergence of the eigenfunction/resolvent expansions without the comparability hypothesis).

Background

The Camassa–Holm (CH) equation $m_t + (um)_x + u_x m = 0$, $m = u - u_{xx}$, derived in 1993 by Camassa and Holm as a model for shallow-water waves, admits weak solutions of peaked-soliton type (peakons), obtained by taking $m$ to be a finite discrete measure. Beals, Sattinger and Szmigielski solved the resulting finite-dimensional peakon system by recognizing the CH equation as an isospectral deformation — a Lax-pair time evolution that preserves the spectrum of an associated boundary value problem — of the classical inhomogeneous string problem $-f'' = \lambda \rho f$ with a measure mass density, a problem studied since the 1950s by M. G. Krein. In the scalar string case, the inverse problem of recovering the mass measure from spectral data is classical Krein theory; in the discrete case it reduces to Stieltjes' theory of continued fractions and the classical moment problem (a survey of this string density problem and its CH applications is given by Eckhardt, Kostenko and Teschl). The paper by Beals and Szmigielski lifted this program from strings (second order) to beams (fourth order). The Euler–Bernoulli beam equation $D^2[r\,D^2\varphi] = \lambda^2 m\varphi$ on $-1 \le x \le 1$, where $r$ is the flexural rigidity (the beam's stiffness against bending), $m$ the mass density, and $D$ the derivative (taken in the sense of distributions when the coefficients are measures), is equivalent, upon setting $n = 1/r$ and writing $\varphi = (\varphi_1, \varphi_2)$, to the matrix system

$$D^2\varphi = \lambda M\varphi, \qquad M = \begin{pmatrix} 0 & n \\ m & 0 \end{pmatrix},$$

that is, an inhomogeneous string equation with a $2\times 2$ matrix-valued measure density. The paper assumes $m$ and $n$ are finite positive measures on $[-1,1]$ with equal support and no atoms at the endpoints, imposes Dirichlet boundary conditions $\varphi_1(\pm 1, \lambda) = \varphi_2(\pm 1, \lambda) = 0$, and develops the spectral theory of the associated compact operator of trace class (meaning its singular values are summable): the eigenvalues are the zeros of $\Delta(\lambda) = \det\Phi(1,\lambda)$, where $\Phi(x,\lambda)$ is the fundamental matrix solution normalized at $x = -1$ ($\Phi(-1,\lambda) = 0$, $D\Phi(-1,\lambda) = \mathbf{1}$), and the boundary spectral data are encoded in the Weyl function

$$W(\lambda) = \frac{1}{\lambda}\, D\Phi(1,\lambda)\,\Phi(1,\lambda)^{-1}.$$

The matrix beam is the isospectral problem for a new two-component Camassa–Holm-type system $n_t = (un)_x + u_x n + vn$, $m_t = (um)_x + u_x m - vm$ with $u_{xx} - 4u = n + m$, $v_x = n - m$, so the inverse spectral problem for general measures is the gateway to solving that system for arbitrary measure-valued (weak) initial data, exactly as scalar string inverse theory underpins the general measure-data solution theory of CH. For the discrete beam, where $m$ and $n$ are supported at finitely many interior points with positive masses, the paper solves the inverse problem completely: $W$ determines the interval lengths and the matrix masses through a non-commutative generalization of Stieltjes' algorithm — Padé approximants of $W$ with matrix coefficients — Padé approximants being rational functions matching a function's series expansion order by order, and Hankel(-like) determinants being determinants of matrices whose anti-diagonals are constant — computed from structured linear systems built from the coefficients of the asymptotic expansion of $W$ at infinity. Three limitations are explicitly acknowledged in the text. First, nothing is done for general (non-discrete) measures. Second, the $L^2$-norm convergence of the eigenfunction and resolvent-kernel expansions is proved only under the additional comparability hypothesis (assumption (5.17) of the paper): there is a constant $C$ with $m + n \le Cm$ and $m + n \le Cn$ measure-theoretically. Third, the partial-fraction (matrix residue) expansion of the Weyl function, the basis of the inverse solution, is justified only in the finite-spectrum case — the paper states verbatim that it omits a detailed justification of formula (6.20) in the general case. (The appendix also identifies more general Lax-pair coefficient parametrizations, with parameters $p, q$ whose non-vanishing would impose a priori algebraic constraints on the measures such as $p^4(mn)^3 = \text{const}$, and dismisses them; classifying those non-generic cases is a separate research direction, not part of the inverse-problem objective.) The open question is the matrix, two-measure version of Krein's classical inverse string problem, where the commutative moment-problem and continued-fraction machinery that solves the scalar case — and that the paper's discrete solution non-commutatively mimics — is no longer available.

Problem Statement

Solve the inverse spectral problem for the Euler–Bernoulli matrix beam (equivalently, the inhomogeneous matrix string) with Dirichlet boundary conditions on $[-1,1]$ for general measures. Specifically, consider the problem $D^2\varphi = \lambda M\varphi$ with $M = \begin{pmatrix} 0 & n \\ m & 0 \end{pmatrix}$, where $m$ and $n$ are finite positive Borel measures on $[-1,1]$ with equal support and no atoms at $\pm 1$, with boundary conditions $\varphi_1(\pm 1, \lambda) = \varphi_2(\pm 1, \lambda) = 0$; let $\Phi(x,\lambda)$ be the fundamental matrix solution normalized by $\Phi(-1,\lambda) = 0$, $D\Phi(-1,\lambda) = \mathbf{1}$, and let the Weyl function be $W(\lambda) = \frac{1}{\lambda}D\Phi(1,\lambda)\Phi(1,\lambda)^{-1}$. Determine:

1. (Characterization) Necessary and sufficient conditions on a $2\times 2$ matrix function $W$ — analyticity structure, asymptotic behavior at infinity and at the origin, and positivity or interlacing-type conditions on the residues — for $W$ to be the Weyl function of such a pair of measures $(m, n)$.
2. (Uniqueness) Whether $W$ determines the pair $(m, n)$, up to a natural equivalence class to be identified if any: either prove injectivity of the map $(m,n) \mapsto W$ on the stated class, or exhibit explicit inequivalent pairs of measures with the same Weyl function and describe the full non-uniqueness.
3. (Reconstruction) A constructive procedure recovering the measures $m$ and $n$ from $W$ up to the natural equivalence class identified in item 2, for general, not necessarily discrete, measures — extending the non-commutative Stieltjes continued-fraction and Padé–Hankel solution that the source paper gives for beams with finitely many point masses.
4. (Spectral foundations) The rigorous justification, for infinite spectrum, of the partial-fraction (matrix residue) expansion of the Weyl function — formula (6.20) of the source, justified there only in the finite-spectrum case — and the convergence properties (weak versus $L^2$) of the associated eigenfunction and resolvent-kernel expansions, removing where possible the comparability hypothesis $m + n \le Cm$, $m + n \le Cn$ under which the currently known $L^2$ convergence is proved.

Items 1–4 form one coherent objective — completing the direct and inverse spectral theory of the matrix beam for general measures — of which item 4 is the analytic backbone for items 1–3, as the source paper's discrete treatment demonstrates; a resolution restricted to discrete or otherwise special measures does not answer the question.

References

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