My main contribution here is the development of an efficient approximation technique for eigenvalue surfaces near multiple eigenvalues – diabolical and exceptional points – and its application to the analytical computation of the geometric phase around exceptional points in non-Hermitian problems of classical and quantum physics. I proposed experimental verification of these findings to the group of Prof. A. Richter at TU-Darmstadt which resulted in our joint work published in Phys. Rev. Lett. in 2011 [5]. The series of these works on non-Hermitian Hamiltonians [1-4] is now actively cited in both physical and mathematical communities.
[1] A.P. Seyranian, O.N. Kirillov, A.A. Mailybaev (2005) Coupling of eigenvalues of complex matrices at diabolic and exceptional points. Journal of Physics A: Mathematical and General. 38(8): 1723-1740
[2] O.N. Kirillov, A.A. Mailybaev, A.P. Seyranian (2005) Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation. Journal of Physics A: Mathematical and General, 38(24): 5531-5546
[3] A.A. Mailybaev, O.N. Kirillov, A.P. Seyranian (2005) Geometric phase around exceptional points. Physical Review A, 72: 014104
[4] A.A. Mailybaev, O.N. Kirillov, A.P. Seyranian (2006) Berry phase around degeneracies. Doklady Mathematics, 73(1): 129-133
[5] B. Dietz, H. L. Harney, O.N. Kirillov, M. Miski-Oglu, A. Richter, F. Schaefer (2011) Exceptional Points in a Microwave Billiard with Time-Reversal Invariance Violation. Physical Review Letters, 106(15): 150403
Dr. Oleg N. Kirillov 