On the M. Kac problem with augmented data
M. I. Belishev,
A. F. Vakulenko St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
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
${\mathbf l}_2$, 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.
Key words and phrases:
M. Kac problem, augmented data, lattice theory, dynamical system with boundary control.
UDC:
517 Received: 06.08.2024