We know for two vector bundles $E$ and $F$ over compact manifold $M$,an elliptic operator $D:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$ is automatically Fredholm.

And for the case $M$ is noncompact, in particular manifolds with cylindrical ends,in this paper: http://www.numdam.org/article/ASNSP_1985_4_12_3_409_0.pdf

Lockhart and Owen introduce the notion weighted Soblev spaces and show that, after putting a suitable weighted Soblev space structure on $\Gamma(\mathrm{E})$ and $\Gamma(\mathrm{F})$, the translation invariant elliptic operators $D_{inv}:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$ are still fredholm, it’s also true even we slightly perturbe $D_{inv}$.

However this beautiful result by Lockhart and Owen requires some assumptions on operator itself.I wonder for any elliptic operator $D:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$, is it possible to put a suitable boundary condition on $\Gamma(\mathrm{E})$ and $\Gamma(\mathrm{F})$ such that $D:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$ is fredholm?

We know for two vector bundles $E$ and $F$ over compact manifold $M$,an elliptic operator $D:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$ is automatically Fredholm.

And for the case $M$ is noncompact, in particular manifolds with cylindrical ends,in this paper: http://www.numdam.org/article/ASNSP_1985_4_12_3_409_0.pdf

Lockhart and McOwens introduce the notion weighted Soblev spaces and show that, after putting a suitable weighted Soblev space structure on $\Gamma(\mathrm{E})$ and $\Gamma(\mathrm{F})$, the translation invariant elliptic operators $D_{inv}:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$ are still fredholm, it’s also true even we slightly perturbe $D_{inv}$.

However this beautiful result by Lockhart and Owen requires some assumptions on operator itself.I wonder for any elliptic operator $D:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$, is it possible to put a suitable boundary condition on $\Gamma(\mathrm{E})$ and $\Gamma(\mathrm{F})$ such that $D:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$ is fredholm?