On the M. Kac problem with augmented data

M. I. Belishev ; A. F. Vakulenko

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 51, Tome 536 (2024), pp. 79-95

Résumé

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 ${\mathbf l}_2$, where $\Phi\colon L_2(\Omega)\to{\mathbf l}_2$ is the Fourier transform. Let $\mathscr K=\{h\in L_2(\Omega) | \Delta h=0 {\rm\ into\ } \Omega\}$ be the harmonic function subspace, $\tilde{\mathscr K}=\Phi\mathscr K\subset{\mathbf l}_2$. 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 $\tilde{\mathscr K}\subset{\mathbf l}_2$ augments the spectrum, making the problem uniquely solvable.