A new representation of (real and virtual) localized modes in the solid state problems is formulated by considering P(z), the modified resolvent operator. It has singularities on the real axis of the z plane only. The isolated poles correspond to real localized modes. When P(E+iη) is analytically continuated onto the second Riemann surface (η > 0), its complex poles correspond to virtual (or unstable) localized modes. The real part of the complex pole is the energy of an unstable localized mode, and the reciprocal of the imaginary part is its life-time. Naturally the life-time is positive. The energy and life-time derived in this way are in accordance with the energy and width of the corresponding resonance scattering. The relation between real and virtual localized modes is discussed. Finally, the singularities of P(z) are related with the change in density of state. The differences between solutions with positive life-times and that with negative life-times are further explained. It is pointed out that, in general, only solutions with positive life-times can change into localized modes when the interaction strength increases, while for solutions with a negative life-time this does not happen.
下载:
