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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2024 Volume 536, Pages 79–95 (Mi znsl7505)

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



© Steklov Math. Inst. of RAS, 2025