Nonlinear dynamics and nuclear collective motion

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To understand the exceedingly rich structure of the quantum excited states in nuclei, the importance of exploring the complex structure of the time-dependent Hartree—Fock (TDHF) manifold is discussed. It is shown that various ideas developed in the general theory of non-linear dynamics (e.g., non-linear resonance, elliptic and hyperbolic fixed points, order-to-chaos transition, etc.,) play a decisive role in obtaining analytic information on the quantum excited states, provided that the corresponding TDHF manifold has a simple potential energy surface (PES) with only one minimum. When the TDHF manifold has a PES with more than two local minima, one is involved into an important problem related with adiabatic versus diabatic collective potentials. The adiabatic collective potential is usually obtained when one numerically solves the constrained Hartree-Fock (CHF) equation within a constraining coordinate space, which has a limited number of degrees of freedom. To explore the dynamical relation between the adiabatic and diabatic single-particle states, one has to analyze the CHF method within the full TDHF manifold, which includes the constraint coordinate space. It turns out that the solutions of the CHF equation give many differentiable surfaces in the TDHF manifold. By using the differentiable property of the CHF solutions in the TDHF manifold, a new method for reaching various HF points is discussed.