  We demonstrate how to design various nonstandard types of Andreev-bound-state
(ABS) dispersions, via a composite construction relying on Majorana bound
states (MBSs). Here, the MBSs appear at the interface of a Josephson junction
consisting of two topological superconductors (TSCs). Each TSC harbors multiple
MBSs per edge by virtue of a chiral or unitary symmetry. We find that, while
the ABS dispersions are $2\pi$-periodic, they still contain multiple crossings
which are protected by the conservation of fermion parity. A single junction
with four interface MBSs and all MBS couplings fully controllable, or, networks
of such coupled junctions with partial coupling tunability, open the door for
topological bandstructures with Weyl points or nodes in synthetic dimensions,
which in turn allow for fermion-parity (FP) pumping with a cycle set by the
ABS-dispersion details. In fact, in the case of nodes, the FP pumping is a
manifestation of chiral anomaly in 2D synthetic spacetime. The possible
experimental demonstration of ABS engineering in these devices, further
promises to unveil new paths for the detection of MBSs and higher-dimensional
chiral anomaly.
