In the document Heat kernel expansion, user's manual Vassilevich expresses in equations $(5.29-5.33)$ the coefficients of the heat kernel expansion up to the fourth one for a manifold with boundaries. It is stated that one recovers Dirichlet and Neuman boundary conditions by imposing either $\Pi_+=0$ or $\Pi_-=0$ respectively since the projectors $\Pi_\alpha$ are defined as: \begin{align} \Pi_- \phi|_{\partial \mathcal{M}}&=0, \\ (\nabla_\textbf{n}+S)\Pi_+ \phi_{\partial \mathcal{M}} &=0. \end{align} With $\nabla_\textbf{n} \equiv n^\mu \nabla_\mu$ for the covariant derivative along the inward pointing (we are in Euclidean space) normal vector $n$ of the boundary, and for an arbitrary function $S$.
My question is then: What about no particular boundary conditions? Should I impose $\Pi_+=\Pi_-=0$? Clearly, some of the terms in $(5.33)$ for example will survive this imposition, so I wonder if this is the right choice to make.
If you want a self-adjoint operator on the manifold (so that you get a discrete spectrum and a heat kernel expansion) then you need exactly one condition at each point of the boundary. There are more options than just Neumann and Dirichlet, you can also take a nontrivial linear combination of the two and you can make different choices at different points of the boundary but you need exactly one condition at each point of the boundary.
If you just look at the unit interval with the second derivative as your manifold you can already see what goes wrong if you have no boundary conditions (spectrum is the entire real line) or if you impose both Dirichlet and Neumann at once (there are no eigenfunctions).