Inverse Spectral Recovery of Vibrating-String Damping

Definitions and mathematical statement

Let $a=a(x)$ be the spatial damping coefficient in the vibrating-string equation \[ u_{tt}(x,t)=u_{xx}(x,t)-2a(x)u_t(x,t),\qquad 0<x<1, \] with homogeneous Dirichlet boundary conditions \[ u(0,t)=u(1,t)=0 \] for all times for which the evolution is considered. The supplied record does not specify the admissible regularity, range, or sign of $a$.

Under the temporal-mode interpretation of the spectrum, substituting $u(x,t)=e^{\lambda t}\phi(x)$, with $\lambda\in{\mathbb{C}}$, gives the quadratic spectral boundary-value problem \[ \phi''(x)-\left(\lambda^2+2\lambda a(x)\right)\phi(x)=0, \qquad \phi(0)=\phi(1)=0,\qquad \phi\not\equiv0. \] Write $\Sigma(a)$ for the spectral data associated with this problem, once the coefficient class, solution spaces, and treatment of multiplicities have been specified. This temporal-mode interpretation makes the spectral question explicit; the supplied record does not itself specify an operator realization or the precise spectral data.

Inverse spectral question. Can $a(x)$ be determined from $\Sigma(a)$? In a uniqueness formulation, after an admissible class $\mathcal A$ and a notion of equality of coefficients have been fixed, the question is whether \[ \forall a_1,a_2\in\mathcal A,\qquad \Sigma(a_1)=\Sigma(a_2)\ \Longrightarrow\ a_1=a_2. \] Here $\mathcal A$ is a placeholder for missing hypotheses, not an additional assumption supplied by the source. Whether recovery means uniqueness, a constructive reconstruction procedure, or recovery up to an equivalence such as reflection of the interval is unresolved in the supplied formulation. Consequently, the record does not determine a single fully specified inverse problem.

Short English statement

Determine whether the spectrum of a string with fixed endpoints determines its spatial damping coefficient. The admissible damping class, precise spectral data, and meaning of recovery need clarification. The supplied record labels the problem partially solved; this is a dated curation assessment, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30001318$, title Inverse Spectral Recovery of Vibrating-String Damping, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2009.
• Supplied original-report citation: Linear and Nonlinear Eigenproblems for PDEs (2010), Oberwolfach Reports, pp. 2025--2114, DOI 10.4171/owr/2009/37. https://doi.org/10.4171/owr/2009/37. The literature-source label is "original Oberwolfach report 4080."
• Supplied literature source, labeled "Bao-Xu-Zhai inverse problem": https://doi.org/10.1137/20M1358827.
• Supplied literature source, labeled "damped-wave inverse spectral theorem": https://arxiv.org/abs/0905.3242.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem partially solved and states that inverse spectral work recovers damping under important restrictions and provides asymptotic or numerical reconstruction algorithms, while the unrestricted vibrating-string question in the report is not completely settled. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.