The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function a on the unit circle from the spectrum of the operator aΛ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Ζm(a) = Τr[(aΛ)2m - (aD)2m] for every smooth function a. In the case of a positive a, zeta-invariants are determined by the Steklov spectrum. The speaker and his collaborator obtain some estimates from below for Ζm(a) in the case of a real function a. On using the estimates, they prove the compactness of a Steklov isospectral family of planar domains in the C∞-topology. They also describe all real functions a satisfying Ζm(a) = 0. The talk is based on a joint work with Alexandre Jollivet.
About the speaker
Prof Vladimir Sharafutdinov obtained his PhD in Differential Geometry at Novosibirsk State University in 1974 and his Doctor of Sciences at Sobolev Institute of Mathematics in 1990. During 1973 – 1977, he was an Assistant Professor at the Novosibirsk State Pedagogical College. He then moved to the Novosibirsk State University in 1978 and is currently a Professor there. He is also the Leading Researcher at the Sobolev Institute of Mathematics in Novosibirsk.
Prof Sharafutdinov's research interests lie in Riemannian geometry, inverse problems and mathematical problems of tomography.
