On the M.Kac problem with augmented data

TL;DR

This paper shows that augmenting the Dirichlet Laplacian spectrum with the harmonic subspace data uniquely determines a domain or manifold, solving a generalized M. Kac problem.

Contribution

The paper demonstrates that the pair of the spectrum and the harmonic subspace data uniquely determines domains and manifolds, extending the classical problem.

Findings

The spectrum plus harmonic subspace data determines the domain up to isometry.

This augmentation solves the inverse problem for Riemannian manifolds.

The result applies to domains and manifolds of arbitrary dimension, metric, and topology.

Abstract

Let ${\Omega}$ be a bounded plane domain. As is known, the spectrum $0<\lambda_1<\lambda_2\leqslant\dots$ of its Dirichlet Laplacian $L=-\Delta{\upharpoonright}[H^2({\Omega})\cap H^1_0({\Omega})]$ does not determine ${\Omega}$ (up to isometry). By this, a reasonable version of the M.Kac problem is to augment the spectrum with relevant data that provide the determination. To give the spectrum is to represent $L$ in the form $\tilde L=\Phi L\Phi^*={\rm diag\,}\{\lambda_1,\lambda_2,\dots\}$ in the space ${\bf l}_2$, where $\Phi:L_2({\Omega})\to{\bf l}_2$ is the Fourier transform. Let ${\mathscr K}=\{h\in L_2({\Omega})\,|\,\,\Delta h=0\,\,{\rm in}\,\,{\Omega}\}$ be the harmonic function subspace, $\tilde{\mathscr K}=\Phi{\mathscr K}\subset{\bf l}_2$. We show that, in a generic case, the pair $\tilde L,\tilde {\mathscr K}$ determines ${\Omega}$ up to isometry, what holds not only for the plain domains (drums) but for the compact Riemannian manifolds of arbitrary dimension, metric, and topology. Thus, the subspace $\tilde{\mathscr K}\subset{\bf l}_2$ augments the spectrum, making the problem uniquely solvable.