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We introduce and study a general pseudodifferential calculus for boundaryvalue problems on a class of non-compact manifolds with boundary (so-called Liemanifolds with boundary). This is accomplished by constructing a suitablegeneralization of the Boutet de Monvel calculus for boundary value problems.The data consists of a compact manifold with corners $M$ that is endowed with aLie structure of vector fields $\mathcal{V}$, a so-called Lie manifold. Themanifold $M$ is split into two equal parts $X_{+}$ and $X_{-}$ which intersectin an embedded hypersurface $Y \subset X_{\pm}$. Our goal is to describe atransmission Boutet de Monvel calculus for boundary value problems compatiblewith the structure of Lie manifolds. Starting with the example of $b$-vectorfields, we show that there are two groupoids integrating the Lie structures on$M$ and on $Y$, respectively. These two groupoids form a bibundle (or agroupoid correspondence) and, under some mild assumptions, these groupoids areMorita equivalent. With the help of the bibundle structure and canonicallydefined manifolds with corners, which are blow-ups in particular cases, wedefine a class of Boutet de Monvel type operators. We then define therepresentation homomorphism for these operators and show closedness undercomposition with the help of a representation theorem. Finally, we considerappropriate Fredholm conditions and construct the parametrices for ellipticoperators in the calculus.

Boutet de Monvel operators on Lie manifolds with boundary