Change in Energy Eigenvalues Against Parameters

A topological characterization of energy-band rearrangements against parameters for molecular problems with slow/fast variables comes around to a study of a Dirac equation with a parameter. In this article, the Dirac equation of space-dimension two is studied under both the APS (an abbreviation of Atiyah–Patodi–Singer) and the chiral bag boundary conditions, where the mass is viewed as a parameter ranging over all real numbers. The APS boundary condition requires that eigenstates evaluated on the boundary should belong to the subspace of eigenstates associated with positive or negative eigenvalues for a boundary operator, and the chiral bag boundary condition requires that eigenstates evaluated on the boundary have chiral components related by a unitary operator. The spectral flow for a one-parameter family of operators is the net number of eigenvalues passing through zeros in the positive direction as the parameter runs. It is shown that the spectral flow for the Dirac equation with the APS boundary condition is ±1, depending on the sign of the total angular momentum eigenvalue. A counterpart of the spectral flow in the case of the chiral bag boundary condition is treated as an extension of spectral flow. In addition, discrete symmetry is discussed to explain the pattern of eigenvalues as functions of the parameter.