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This paper concerns the $\mathbb{Z}_2$ classification of FermionicTime-Reversal (FTR) symmetric partial differential Hamiltonians on theEuclidean plane. We consider the setting of two insulators separated by aninterface. Hamiltonians that are invariant with respect to spatial translationsalong the interface are classified into two categories depending on whetherthey may or may not be gapped by continuous deformations. Introducing a relatedodd-symmetric Fredholm operator, we show that the classification is stableagainst FTR-symmetric perturbations.  The property that non-trivial Hamiltonians cannot be gapped may beinterpreted as a topological obstruction to Anderson localization: no matterhow much (spatially compactly supported) perturbations are present in thesystem, a certain amount of transmission in both directions is guaranteed inthe nontrivial phase. We present a scattering theory for such systems and shownumerically that transmission is indeed guaranteed in the presence ofFTR-symmetric perturbations while it no longer is for non-symmetricfluctuations.

$\mathbb{Z}_2$ classification of FTR symmetric differential operators and obstruction to Anderson localization