A logarithm representation of evolution operators is defined. Generators of invertible evolution families are characterized by the logarithm representation. In this article, using the logarithm representation, a concept of evolution operators without satisfying the semigroup property is introduced. In conclusion the existence of alternative infinitesimal generator is clarified.

Charge equilibration between two colliding nuclei can take place in the early stage of heavy-ion collisions. A basic mechanism of charge equilibration is presented in terms of the extension of singleparticle motion from one nucleus to the other, from which the upper energy-limit of the bombarding energy is introduced for significant charge equilibration at the early stage of the collision. The formula for this limit is presented, and is compared to various experimental data. It is examined also by comparison to three-dimensional time-dependent density functional calculations. The suppression of charge equilibration, which appears in collisions at the energies beyond the upper energy-limit, gives rise to remarkable effects on the synthesis of exotic nuclei with extreme proton-neutron asymmetry.PACS numbers: 21.10. Ft, 25.70.Hi Charge equilibration is a rapid process during the early stage of heavy-ion collisions with a time scale of 10 −22 sec (for a review, see [1]). Despite many theoretical attempts, its mechanism has remained an open problem, where only the relation with the isovector giant dipole resonance (iv-GDR) was discussed in certain cases (for example, see [2][3][4][5][6][7][8]). The charge equilibration is quite important, because it naturally prevents the synthesis of exotic nuclei with extreme proton-neutron asymmetry. As there are and will be the 3rd generation RI-beam facilities, it is an urgent question whether such synthesis can be enhanced with higher beam energies or not.In this Letter, we first point out that there is an upper limit of the bombarding energy for the fast and significant charge equilibration in the initial stage of the collision. Microscopic three-dimensional (time-dependent) density functional calculations are systematically performed for the purpose of examination. We actually employ Skyrme time-dependent Hartree-Fock (TDHF) theory, which is a rather unique presently feasible method for the treatment of non-perturbative processes such as multi-nucleon transfers in a realistic framework.The suppression of charge equilibration brings about a favorable situation for the synthesis of exotic nuclei. The evaluation of the upper limit thus can have crucial significance for experiments on nuclei with extreme protonneutron asymmetry. We shall consider the collision of a target nucleus with mass number A 1 , neutron (proton) number N 1 (Z 1 ), with a projectile nucleus with A 2 , N 2 , and Z 2 . The total mass, neutron, and proton numbers are denoted by A, N , Z, respectively. We look at this problem from an intuitive and basic viewpoint. We begin with a picture that the charge equilibration takes place as wave functions of nucleons propagate from their original nucleus to the other nucleus in the initial stage of the collision. Namely, the regime of individual singleparticle motion spreads out following the lowering of potential barrier between the two nuclei after the touching.Because this spreading occurs as a consequence of unblocked single-particle motion, the process can be ve...

We present the nuclear matrix element for the neutrinoless double-beta decay of ^{48}Ca based on large-scale shell-model calculations including two harmonic oscillator shells (sd and pf shells). The excitation spectra of ^{48}Ca and ^{48}Ti, and the two-neutrino double-beta decay of ^{48}Ca are reproduced in good agreement to the experimental data. We find that the neutrinoless double-beta decay nuclear matrix element is enhanced by about 30% compared to pf-shell calculations. This reduces the decay lifetime by almost a factor of 2. The matrix-element increase is mostly due to pairing correlations associated with cross-shell sd-pf excitations. We also investigate possible implications for heavier neutrinoless double-beta decay candidates.

The tensor and spin-orbit forces contribute essentially to the formation of the spin mean field, and give rise to the same dynamical effect, namely spin polarization. In this paper, based on timedependent density functional calculations, we show that the tensor force, which usually acts like a small correction to the spin-orbit force, becomes more important in heavy-ion reactions and the effect increases with the mass of the system.