Condition for emergence of complex eigenvalues in the Bogoliubov-de Gennes equations

Nakamura, Yusuke*; Mine, Makoto*; Okumura, Masahiko   ; Yamanaka, Yoshiya*

The condition for the appearance of dynamical instability of the Bose- condensed system, characterized by the emergence of complex eigenvalues in the Bogoliubov-de Gennes equations, is studied analytically. It is concluded that the degeneracy between a positive- norm eigenmode and a negative-norm one is essential for the emergence of complex modes. Based on the conclusion, we justify the two-mode approximation applied in our previous work, in which we analytically studied the condition for the existence of complex modes when the condensate has a highly quantized vortex.