An inverse problem for a general convex domain with impedance boundary conditions

The spectral function $\theta \left ( t \right ) = \sum \nolimits _{n = 1}^\infty {\exp \left ( { - t{\lambda _n}} \right )}$, where $\left \{ {{\lambda _n}} \right \}_{n = 1}^\infty$ are the eigenvalues of the Laplace operator $\Delta = \sum \nolimits _{i = 1}^2 {{{\left ( {\partial /\partial {x^i}} \right )}^2}}$ in the ${x^1}{x^2}$-plane, is studied for a general convex domain $\Omega \subseteq {R^2}$ with a smooth boundary $\partial \Omega$ together with a finite number of piecewise smooth impedance boundary conditions on the parts ${\Gamma _{1,...,}}{\Gamma _m}$ of $\partial \Omega$ such that $\partial \Omega = U_{j = 1}^m{\Gamma _j}$.