N ov 2 00 4 The hybrid spectral problem and Robin boundary conditions

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C 1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semicircles , is derived. Comments on higher coefficients are made. A separable hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat– kernel expansion which are evaluated explicitly.