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We theoretically analyze a discrete Schr\"odinger chain with hopping to thefirst and second neighbors, as can be realized with zigzag arrangements ofoptical waveguides or lattice sites for cold atoms. Already at moderate values,the second-neighbor hopping has a strong impact on the band structure, leadingto the emergence of a new extremum located inside the band, accompanied by avan Hove singularity in the density of states. The energy band is then dividedinto a subcritical regime with the usual unique correspondence between wavenumber and energy of the travelling waves, and a supercritical regime, in whichwaves of different wave number are degenerate in energy. We study theconsequences of these features in a scattering setup, introducing a defect thatlocally breaks the translational invariance. The notion of a local probabilitycurrent is generalized beyond the nearest-neighbor approximation and boundstates with energies outside the band are discussed. At subcritical energiesinside the band, an evanescent mode coexists with the travelling plane wave,giving rise to resonance phenomena in scattering. At weak coupling to thedefect, we identify a prototypical Fano-Feshbach resonance of tunable shape andprovide analytical expressions for its profile parameters. At supercriticalenergies, we observe coupling of the degenerate travelling waves, leading to anintricate wave packet fragmentation dynamics. The corresponding branchingratios are analyzed.

Sub- and supercritical defect scattering in Schr\"odinger chains with higher-order hopping