XXXIV Workshop on Geometric Methods in Physics 28.06-4.07.2015
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# Toshihiro Iwai

## Change in energy eigenvalues against parameters

The Dirac equations of space-dimension two is studied under the APS boundary condition,where the mass is considered as a parameter ranging over all real numbers and where APS is an abbreviation of Atiyah-Patodi-Singer and the boundary condition requires that the boundary values of eigenstates of the Dirac equation should belong to the subspace of eigenstates associated with positive or negative eigenvalues for a boundary 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 demonstrated that the spectral flow for the Dirac equation with the APS boundary condition is $\pm$, depending on the sign of the total angular momentum eigenvalue.
The chiral bag boundary condition is also treated as comparison's sake.
In addition, discrete symmetry for the associated energy bands is discussed.
Related topics will be touched upon, including winding numbers for the corresponding semi-quantum" Hamiltonian, rotation-vibration systems in quantum chemistry for isolated molecules, comparison with topological insulators, and further extension of the present model.
This talk is based on joint works with B. Zhilinskii at Universit{\' e} du Littoral C\^{o}te d'Opale.

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