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Rearrangement of energy bands against a parameter is studied through the3D-Dirac equation on a ball in $\mathbb{R}^3$ under the APS and the chiral bagboundary conditions on the boundary two-sphere, where APS is an abbreviation ofAtiyah-Patodi-Singer. The notion of spectral flow and its extension isintroduced to characterize the energy eigenvalue redistribution against theparameter, which reflects an analytical property of edge eigenstates. It isshown that though band rearrangement takes place the net spectral flow is zero($1+(-1)=0$) for both boundary conditions. The corresponding semi-quantumHamiltonian defined on the 3D momentum space is studied in parallel and it isshown that a change against the parameter is observed in the mapping degreedefined for the semi-quantum model Hamiltonian, which is viewed as reflecting atopological property of the bulk eigenstates of the full quantum system.Specifically, there are two mappings assigned, for which changes in mappingdegree take the values $\pm1$. The present correspondence is viewed as abulk-edge correspondence.

Band rearrangement through the 3D-Dirac equation with boundary conditions, and the corresponding topological change