Basic and Fluctuating Periodic Instantons in Quantum Tunneling

Under condition of four potential fields, equations of motion and fluctuations in imaginary time are utilized to analytically derive the basic and fluctuating periodic instantons. It is shown that the basic instantons satisfy the elliptic or simple pendulum equations and their solutions are Jacobi elliptic functions, and fluctuating periodic instantons satisfy the Lamé equation and their solutions are Lamé functions. These results indicate that there exists the common solution family for different potential fields which are called the super-symmetry family.