Parametrix Construction and Invertibility of ${\mathrm{\Delta }}_g+1$

Given a manifold $M$ with an asymptotically cylindrical end, we construct the parametrix for the shifted Laplace operator ${\mathrm{\Delta }}_g+1$ on the blown-up $b$-double space $M_b^2$. We will review the geometry of manifold with a cylindrical end and construct the inverse directly via separation of variables, which motivates the parametrix construction. Next, we review the mapping properties and show how to get elliptic regularity from the mapping properties of the parametrix. Through this explicit example, we hope this paper provides another reference for the theory of $b$-operators introduced by Melrose \cite{Mel93}.