/ Kenichi Kuroda / Professor
/ Hiroyuki Sagawa / Professor
/ Noriaki Kamiya / Professor
/ Ken-ichi Funahashi /Associate Professor
/ Katsutaro Shimizu / Associate Professor
/ Kazuto Asai / Assistant Professor
/ Michio Honma / Assistant Professor
/ Shigeru Watanabe / Assistant Professor
/ Toshiro Watanabe / Assistant Professor
/ Hiroshi kihara / Research Associate
The scope of activities of the Center for Mathematical sciences spans all aspects of education and research in the fields of mathematical sciences. Our current researches in the field of mathematics are devoted to various subjects and problems arising in both pure and applied mathematics. In the fields of Physics, theoretical research is performed in many-body theories, Nuclear Physics and Quantum Gravity. There has been also a project to develop educational textbooks and software for mechanics, electromagnetism and quantum mechanics. The research areas assigned to each co-researcher are as follows:
Refereed Journal Papers
We study displacement fields of multipole excitations in a halo nucleus $^{11}_4 $Be$_{7}$ and a skin nucleus $^{60}_{20}$Ca$_{40}$~ in comparison with those of collective states in a $\beta-$stable nucleus $^{208}_{82}$Pb$_{126}$. The self-consistent Hartree-Fock calculation plus the random phase approximation with Skyrme interactions is performed in the coordinate space, taking into account the proton and neutron degree of freedom.
We studied Electromagnetic moments of the $\beta- $emitting nucleus $^{19}$O, experimentally and theoetically.
We have studied dipole states of oxygen-isotopes in large scale shell modelcalculations. The calculated photoreaction cross sections in $^{16}$O and $^{18}$O give reasonable agreement with experimental observations both in the low energy region below $\hbar \omega $=15 MeV and in the high energy giant resonance region (15 MeV $<\hbar \omega \leq $30 MeV).
We have studied electric dipole (E1) and spin-dipole strength distribution in $^{11}$Li by shell model calculations with halo effect. Two peaks in the E1 strength are found in the low excitation energy region below E$_{x}$ = 4 MeV, which have almost the same energies as observed E1 peaks in the Coulomb breakup reaction of $^{11}$Li. The calculated E1 strength up to E$_{x}$ = 4 MeV exhaust about 4 $\%$ of the Thomas-Reiche-Kuhn energy-weighted sum rule value. We found also pigmy and giant peaks in spin-dipole strengths of $^{11}$Li.
Spin-orbit interactions in a proton drip-line nucleus $^{72}$Kr are discussed in Skyrme Hartree-Fock and relativistic mean field theories by using various parameter sets. We found that the two models give different results not only in the spin-orbit splitting but also in the angular momentum dependence of the single-particle energies. The Skyrme force SLy10 in non-relativistic model and NL2, NL3 and NLSH interactions in the relativistic model give the single-particle energy of $1f_{5/2}$ state lower than that of $2p_{3/2}$ state, which agree with recently suggested level ordering in the analysis of $\beta$-decay experiment of $^{71}$Kr.
The response of light drip line nuclei to spin-isospin dependent fields is investigated, using the self-consistent Hartree-Fock plus the random-phase approximation with Skyrme interactions. Including simultaneously both the isoscalar and isovector spin correlation, the RPA response function is estimated in the coordinate space so as to take properly into account the continuum effect. The spin-orbit splitting, which plays an essential role in the spin-dependent response functions, is examined as a function of the mass number when we approach the drip line nuclei. It is found that the calculated M1 peaks in $^{8}_{6}$C$_{2}$ and $^{8}_{2}$He$_{6}$ are much lower in energy than those expected from our knowledge of the $\beta$-stable nucleus $^{12}_{6}$C$_{6}$.
The structure of low-lying states of $^{56}$Ni is studied by the Monte Carlo shell model based on the quantum Monte Carlo diagonalization method. The coexistence of spherical yrast, prolate deformed, and other non-yrast states is described by the full $pf$-shell calculations, by employing the FPD6 realistic residual interaction. To understand the properties of eigenstates thus obtained, we utilize a mean field analysis, such as a potential energy surface by constrained Hartree-Fock method.
We propose a quantum Monte Carlo diagonalization method (QMCD) for solving the quantum many-body interacting systems. Not only the ground state but also low-lying excited states are obtained with their wavefunctions. Consequently the level structure of low-lying states can be studied with realistic interactions. The developments in the formulation of the QMCD are described with an illustrative and intuitive example. The QMCD is finally characterized as a 'importance truncation' scheme to the shell model. After testing this method for $^{48}$Cr, we present first results for energy levels and E2 properties of $^{64}$Ge, indicating its large and $\gamma$-soft deformation. The doubly closed shell probability of $^{56}$Ni is shown to be only 49\% in a full pf shell calculation, in contrast to the corresponding probability of $^{48}$Ca which reaches 86\%. The prolate deformed excited band is obtained as a result of the QMCD calculation for non-yrast states, in a good agreement to recent experimental data. This band seems to have an $SU$(3) (-like) structure which was originally suggested by Elliott for sd-shell nuclei.
The structure of neutron-rich nuclei in the $N\sim 20$ region is studied by the Monte Carlo shell model based on the quantum Monte Carlo diagonalization method. We present a comprehensive description of even-even isotopes of O, Ne, Mg, and Si. It is demonstrated that, for different nuclei, various particle-hole excitations from the $sd$ to $pf$ are mixed in different ways, producing distinct effects sometimes, for instance, in $^{28}$Ne. The monopole interaction is examined and modified, resulting in the shell gap changing from nucleus to nucleus. The drip line of O isotopes is then reproduced. The interplay between the $T$=0 and $T$=1 monopole interactions is discussed from the viewpoint of the potential energy surface and the effective single-particle energy. The extension of the neutron drip line of Ne isotopes is explained, and the boundary of the 'island of inversion' is shown to be rather indistinct.
This paper is an application to Yang-Baxter equations from Lie algebras.
In this paper, we have investigated about radicals for Freudenthal-Kantor triple systems.
This paper is a construction of simple superalgebras from triple systems.
In this paper, we have investigated a correspondence of Lie supertriple systems with Freudenthal-Kantor supertriple systems.
P.S. numbers are introduced in relation to absolute continuity of some infinite Bernoulli convolutions. Absolute continuity and continuous singularity of some semi-selfdecomposable distributions are studied as marginal distributions of subordinators. It is shown that these properties are widely different according as their spans are P.V. numbers or the reciprocals of P.S. numbers. A simple example of a subordinator whose distribution is continuous singular for small time and absolutely continuous for large time is given. Absolute continuity of convolutions of multidimensional homogeneous self-similar measures are also discussed.
The exponent of a semi-selfsimilar process is shown to exist under the mere assumption of stochastic continuity at $t = 0$, and related examples are given. A relationship between long range dependence of the increments and the exponent is also discussed.
The class of completely operator semi-selfdecomposable distributions are characterized. The fact that this class is the smallest operator-completely closed class in the strong sense containing the class of operator-semistable distributions is shown. It is a semi-version of Sato-Yamazato's theorem.
Relationships between marginal and joint distributions of selfsimilar processes with independent increments are shown in terms of the Urbanik-Sato-type nested subclasses of the class of delfdecomposable distributions. Similar results are also shown for semi-selfsimilar processes with independent increments.
In the previous paper we showed a generating function for the zonal spherical functions on the homogeneous space $U(n)/U(n-1)$ can be regarded as the integral kernel of a unitary mapping from an $L^2$ space onto a Hilbert space of analytic functions. The purpose of the present paper is to give some characterization of the zonal spherical functions on $U(n)/U(n-1)$ by using results in the previous paper.
Refereed Proceeding Papers
Electric dipole states in light neutron-rich nuclei are studied by large scale shell model calculations. The calculated photoreaction cross sections in oxygen-isotopes and $^{14}$C give reasonable agreement with experimental observations both in thelow energy region below $\hbar \omega $=15 MeV and in the high energy giant resonance region (15 MeV $<\hbar \omega \leq $30 MeV). We found that the transition strength below dipole giant resonance ($\hbar \omega \leq $15 MeV) exhausts about 10\% of the classical Thomas-Reiche-Kuhn sum rule value in heavier oxygen isotopes than $^{18}$O, which is consistent with new experimental data from GSI.
We have studied dipole states of oxygen-isotopes in large scale shell model calculations. The calculated photoreaction cross sections in $^{16}$O, $^{17}$O and $^{18}$O give reasonable agreement with experimental observations both in the low energy region below $\hbar \omega $=15 MeV and in the high energy giant resonance region (15 MeV $<\hbar \omega \leq $30 MeV). We found that the transition strength below dipole giant resonance ($\hbar \omega \leq $15 MeV) exhausts about 10\% of the classical Thomas-Reiche-Kuhn sum rule and more than 40\% of the cluster sum rule in heavier oxygen isotopes than $^{17}$O. The predicted Pigmy strengths in $^{20}$O and $^{22}$O are also confirmed by recent Coulomb excitation experiment at GSI.
The yrast structure of even-even Fe isotopes with masses $A=52,54,56,58,60$ are studied on the basis of the shell model by using the Quantum Monte Carlo diagonalization (QMCD) method. The realistic shell model Hamiltonian is diagonalized approximately in the complete pf-shell by selecting dominant configurations stochastically. Calculated energy levels and transition properties are in good agreement with the experimental data for low-lying states. Yrast quasi-rotational band with prolate deformation are found in low-lying states of $^{52}$Fe, $^{56}$Fe and $^{60}$Fe. In $^{60}$Fe, an oblate band becomes yrast at high-spin($J\sim$ 10). The intrinsic states which describes the shell model states are considered by using dominant components of the QMCD basis. The deformation parameters are derived from the intrinsic states, which show clear correspondence with the band structure.
Let $O$ be a finite odd-ary tree as a digraph or equivalently, as an ordered set. Let $K=(K_1,\dots,K_r)$ be connected induced subdigraphs of $O$, including all the edges incident with ramification vertices of $O$. We deal with tableaux of order ideals of $K_1,\dots,K_r$, with $r$ rows and infinitely many columns. The $(i,j)$-entry $T_{ij}$ of the tableau is an order ideal of $K_i$, which in creases as $j$ increases, and satisfies $T_{ij}\cap K_{i+1+l}\subset T_{i+1+l,j+l}\ cap K_i$. We define a certain class ${\rm Tab}(K,B,\alpha)$ of the above tableaux and construct the bijection between that class and a set of ``tree-like'' disjoint paths in a special digraph. In virtue of this, together with an odd-ary-tree-analogue of Lindstr\"om's theorem, we prove the Jacobi--Trudi identity for ${\rm Tab}(K,B, \alpha)$ with a superdeterminant of even dimension $s=\#\{\text{end vertices of }O\}$.
Books
We have described about triple systems, in particular, Freudenthal-Kantor triple systems, Lie triple systems, etc.
Academic Activities