There is indeed a pseudodifferential calculus that can handle this situation. It is called SG calculus, see e.g. here, or scattering calculus, see e.g. here. Symbol estimates (on $\mathbb R^N$) are $$ |\partial_x^\alpha \partial_\xi^\beta a(x,\xi)| \leq C_{\alpha\beta} (1+|x|)^{p-|\alpha|} (1+|\xi|)^{m-|\beta|}. $$ This calculus can be introduced on manifolds $\boldsymbol X$ with cylindrical ends (also called SG or scattering manifolds). Pseudodifferential operators act between weighted function spaces. The Fredholm property of an operator $A$ is equivalent to its ellipticity, where the latter means (for a classical operator $A$) pointwise invertibility of the triplet $$ (\sigma_\psi^m(A),\sigma_e^p(A),\sigma_{e,\psi}^{p,m}(A)) $$ (i.e., it is in fact ellipticity in a calculus with symbolic structure). In case $X=\mathbb R^N$ and $A=a(x,D)$,
$\sigma_\psi^m(A)(x,\xi) = \lim_{\lambda\to\infty} \lambda^{-m} a(x,\lambda \xi)\,$ for $(x,\xi)\in \mathbb R^N\times(\mathbb R^N\setminus0)$ is the usual principal symbol,
$\sigma_e^p(A)(x,\xi) = \lim_{\lambda\to\infty}\mu^{-p} a(\mu x,\xi)\,$ for $(x,\xi)\in (\mathbb R^N\setminus0)\times\mathbb R^N$ is the principal "exit" symbol,
$\sigma_{e,\psi}^{p,m}(A)$ arises as a compatibility condition between the former two: $$ \sigma_{e,\psi}^{p,m}(A)(x,\xi) = \lim_{\mu\to\infty} \mu^{-p} \sigma_\psi^m(A)(\mu x,\xi) = \lim_{\lambda\to\infty} \lambda^{-m} \sigma_e^p(A)(x,\lambda \xi), $$ $(x,\xi)\in (\mathbb R^N\setminus0)\times(\mathbb R^N\setminus0)$.
In your example, the operator $1-\Delta$ is SG elliptic, while the operator $-\Delta$ is not.