Examples of Static Localized Solutions of Nonlinear Equations in 3+1 Dimensions. III ---Bound Mode with an Imaginary Frequency---

There often appears a fluctuation with an imaginary frequency in the theory of the extended object. For the first time, a method is proposed to solve such an imaginary frequency mode by making use of the Yang-Feldman equation. It is found that the presence of such a mode does not influence the energy of the extended object. The nonlinear equation, for which this imaginary frequency mode obeys, further contains infinitely many other solutions with a continuous range of imaginary frequencies. These solutions are acceptable mathematically but have to be rejected in quantum mechanics beacuse they diverge at infinity. As examples, we will solve two of these solutions having small frequencies.