The basic problem is to determine the geometry of an arbitrary multiply connected bounded region in R 2 together with the mixed boundary conditions, from the complete knowledge of the eigenvalues { λ i } j = 1 ∞ for the Laplace operator, using the asymptotic expansion of the spectral function θ ( t ) = ∑ j = 1 ∞ exp ( − t λ i ) as t → 0 .