On the M.Kac problem with augmented data

Let Ω be a bounded plane domain. As is known, the spectrum 0<λ₁<λ₂ of its Dirichlet Laplacian L=-Δ[H²(Ω)∩ H¹₀(Ω)] does not determine Ω (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 L=Φ LΦ^*= diag\,\λ₁,λ₂,\ in the space l₂, where Φ:L₂(Ω)→ l₂ is the Fourier transform. Let K=∈ L₂(Ω)\,|\,\,Δ h=0\,\, in\,\,Ω\ be the harmonic function subspace, K̃=Φ K⊂ l₂. We show that, in a generic case, the pair L, K̃ determines Ω 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 K̃⊂ l₂ augments the spectrum, making the problem uniquely solvable.