Takayanagi Kazuo

Abstract We present the generalized optical theorem and its applications with special emphasis on the roles of bound states. First, we prove the theorem which gives a necessary and sufficient condition for a function $\langle {\boldsymbol {k } }^{\prime } | T | {\boldsymbol {k } } \rangle$ of two variables ${\boldsymbol {k } }^{\prime }$ and ${\boldsymbol {k } }$ to be physically acceptable as a half-on-shell T-matrix, i.e., to have an underlying Hermitian potential V. Secondly, using the theorem, we construct a scattering theory starting from a physically acceptable half-on-shell T-matrix $\langle {\boldsymbol {k } }^{\prime } | T | {\boldsymbol {k } } \rangle$, which in turn introduces a very useful classification scheme of Hermitian potentials. In the end, as an application of our theory, we present the most general solution of the inverse scattering problem with numerical examples.

Atomic nuclei are composed of a certain number of protons Z and neutrons N. A natural question is how large Z and N can be. The study of superheavy elements explores the large Z limit(1,2), and we are still looking for a comprehensive theoretical explanation of the largest possible N for a given Z-the existence limit for the neutron-rich isotopes of a given atomic species, known as the neutron dripline(3). The neutron dripline of oxygen (Z = 8) can be understood theoretically as the result of single nucleons filling single-particle orbits confined by a mean potential, and experiments confirm this interpretation. However, recent experiments on heavier elements are at odds with this description. Here we show that the neutron dripline from fluorine (Z = 9) to magnesium (Z = 12) can be predicted using a mechanism that goes beyond the single-particle picture: as the number of neutrons increases, the nuclear shape assumes an increasingly ellipsoidal deformation, leading to a higher binding energy. The saturation of this effect (when the nucleus cannot be further deformed) yields the neutron dripline: beyond this maximum N, the isotope is unbound and further neutrons 'drip' out when added. Our calculations are based on a recently developed effective nucleon-nucleon interaction(4), for which large-scale eigenvalue problems are solved using configuration-interaction simulations. The results obtained show good agreement with experiments, even for excitation energies of low-lying states, up to the nucleus of magnesium-40 (which has 28 neutrons). The proposed mechanism for the formation of the neutron dripline has the potential to stimulate further thinking in the field towards explaining nucleosynthesis with neutron-rich nuclei.A mechanistic explanation for the origin of the neutron dripline shows that nuclei accommodate the addition of neutrons by becoming increasingly ellipsoidal, up to a maximum number of neutrons, reconciling theory and experiments.

We present the first application of the newly developed extended Kuo-Krenciglowa (EKK) theory of the effective nucleon-nucleon interaction to shell-model studies of exotic nuclei, including those where conventional approaches with fitted interactions encounter difficulties. This EKK theory enables us to derive an interaction that is suitable for several major shells (sd + pf in this work). By using such an effective interaction obtained from the Entem-Machleidt QCD-based chi(NLO)-L-3 interaction and the Fujita-Miyazawa three-body force, the energies, E2 properties, and spectroscopic factors of low-lying states of neutron-rich Ne, Mg, and Si isotopes are nicely described, as the first shell-model description of the "island of inversion" without fit of the interaction. The long-standing question as to how particle-hole excitations occur across the sd-pf magic gap is clarified with distinct differences from the conventional approaches. The shell evolution is shown to appear similarly to earlier studies.

We present a unified description of effective interaction theories in both algebraic and graphic representations. In our previous work, we have presented the Rayleigh-Schrodinger and Bloch perturbation theories in a unified fashion by introducing the main frame expansion of the effective interaction. In this work, we start also from the main frame expansion, and present various nonperturbative theories in a coherent manner, which include generalizations of the Brandow, Brillouin-Wigner, and Bloch-Horowitz theories on the formal side, and the extended Krenciglowa-Kuo and the extended Lee-Suzuki methods on the practical side. We thus establish a coherent and comprehensive description of both perturbative and nonperturbative theories on the basis of the main frame expansion. (C) 2016 Elsevier Inc. All rights reserved.

We present a unified description of the Bloch and Rayleigh-Schrodinger perturbation theories of the effective interaction in both algebraic and graphic representations. (C) 2015 Elsevier Inc. All rights reserved.

We present a novel solution to the inverse scattering problem. Our solution is based on a generalization of the optical theorem, and applies directly in three dimensional space. First, we derive a necessary and sufficient condition for a half-on-shell T-matrix to be physically acceptable, which turns out to be a generalization of the optical theorem. Second, we show that the inverse scattering problem, which inquires underlying potential for a given on-shell T-matrix (scattering amplitude), can be solved by looking for a half-on-shell T-matrix that satisfies the generalized optical theorem with the given on-shell T-matrix being the boundary condition. At the end, we demonstrate that the present theory works nicely using simple systems. (C) 2015 AIP Publishing LLC.

We present a unified description of the effective interaction v in the Rayleigh-Schrodinger perturbation theory. First, we generalize the well-known bracketing expression for the energy shift Delta E in a one-dimensional model space to express the effective interaction upsilon in a multi-dimensional model space. Second, we show that the generalized bracketing representation has a natural graphic expression in terms of folded diagrams. The present work thus gives a unified understanding of the effective interaction (i) in one- and multi-dimensional model spaces and (ii) in algebraic (bracketing) and graphic (folded diagram) representations. (C) 2014 Elsevier Inc. All rights reserved.

Background: Effective interactions, either derived from microscopic theories or based on fitting selected properties of nuclei in specific mass regions, are widely used inputs to shell-model studies of nuclei. The commonly used unperturbed basis functions are given by the harmonic oscillator. Until recently, most shell-model calculations have been confined to a single oscillator shell like the sd shell or the pf shell. Recent interest in nuclei away from the stability line requires, however, larger shell-model spaces. Because the derivation of microscopic effective interactions has been limited to degenerate models spaces, there are both conceptual and practical limits to present shell-model calculations that utilize such interactions. Purpose: The aim of this work is to present a novel microscopic method to calculate effective nucleon-nucleon interactions for the nuclear shell model. Its main difference from existing theories is that it can be applied not only to degenerate model spaces but also to nondegenerate model spaces. This has important consequences, in particular for intershell matrix elements of effective interactions. Methods: The formalism is presented in the form of a many-body perturbation theory based on the recently developed extended Kuo-Krenciglowa method. Our method enables us to microscopically construct effective interactions not only in one oscillator shell but also for several oscillator shells. Results: We present numerical results using effective interactions within (i) a single oscillator shell (a so-called degenerate model space) like the sd shell or the pf shell and (ii) two major shells (nondegenerate model space) like the sdf7p3 shell or the pfg9 shell. We also present energy levels of several nuclei that have two valence nucleons on top of a given closed-shell core. Conclusions: Our results show that the present method works excellently in shell-model spaces that comprise several oscillator shells, as well as in a single oscillator shell. We show, in particular, that the microscopic intershell interactions are much more attractive than has been expected by degenerate perturbation theory. The consequences for shell-model studies are discussed. © 2014 American Physical Society.

Background: Effective interactions, either derived from microscopic theories or based on fitting selected properties of nuclei in specific mass regions, are widely used inputs to shell-model studies of nuclei. The commonly used unperturbed basis functions are given by the harmonic oscillator. Until recently, most shell-model calculations have been confined to a single oscillator shell like the sd shell or the pf shell. Recent interest in nuclei away from the stability line requires, however, larger shell-model spaces. Because the derivation of microscopic effective interactions has been limited to degenerate models spaces, there are both conceptual and practical limits to present shell-model calculations that utilize such interactions. Purpose: The aim of this work is to present a novel microscopic method to calculate effective nucleon-nucleon interactions for the nuclear shell model. Its main difference from existing theories is that it can be applied not only to degenerate model spaces but also to nondegenerate model spaces. This has important consequences, in particular for intershell matrix elements of effective interactions. Methods: The formalism is presented in the form of a many-body perturbation theory based on the recently developed extended Kuo-Krenciglowa method. Our method enables us to microscopically construct effective interactions not only in one oscillator shell but also for several oscillator shells. Results: We present numerical results using effective interactions within (i) a single oscillator shell (a so-called degenerate model space) like the sd shell or the pf shell and (ii) two major shells (nondegenerate model space) like the sdf(7)p(3) shell or the pfg(9) shell. We also present energy levels of several nuclei that have two valence nucleons on top of a given closed-shell core. Conclusions: Our results show that the present method works excellently in shell-model spaces that comprise several oscillator shells, as well as in a single oscillator shell. We show, in particular, that the microscopic intershell interactions are much more attractive than has been expected by degenerate perturbation theory. The consequences for shell-model studies are discussed.

The effective Hamiltonian in a model space has been derived from the decoupling equation. We present a rigorous proof of the proposition that the decoupling equation is a necessary and sufficient condition to give an effective Hamiltonian, establishing a robust one-to-one correspondence between a solution to the decoupling equation and an effective Hamiltonian. We then present discussions based on this result, emphasizing (i) that the current situation of the theory is far from being satisfactory, and (ii) that the proposition gives a rigorous mathematical foundation to any effort to improve the theory of the effective Hamiltonian on the basis of the decoupling equation. © 2013 Elsevier B.V.

We present a novel method to calculate a microscopic effective interaction in many-body systems. This method applies not only to degenerate model spaces but also to non-degenerate model spaces, and therefore allows us to construct microscopically the effective interaction in multi-major-shell model spaces. As an application of this novel method, we examine the effective interaction between valence neutrons of O-18 in the sd-shell (model space of single-major-shell), and in the sdf(7)p(3)-shell (model space of multi-major-shells). We find that the microscopic cross-shell interactions are much more attractive than have been expected by ad hoc methods. Our approach presents an essential step to a microscopic description of nuclei in multi-major-shells, which is crucial, e.g., to describe many of the neutron-rich nuclei.

The extended Krenciglowa-Kuo (EKK) method allows us to calculate the effective Hamiltonian in a non-degenerate model space. We show that the EKK method can be implemented numerically in two iterative schemes, which are explained in detail with emphasis on convergence conditions. Using test calculations in a simple model, we clarify how and on what conditions we can calculate the effective Hamiltonian. (C) 2011 Elsevier B.V. All rights reserved.

The effective interaction in a model space has been calculated by the Krenciglowa-Kuo (KK) and the Lee-Suzuki (LS) iterative methods, both of which assume that the unperturbed energies in the model space are degenerate. We generalize these two methods in a natural and simple manner so that they apply also to non-degenerate model spaces. The key to the generalization is to use the effective hamiltonian instead of the effective interaction in the formulation of iterative schemes. Using test calculations in a simple model, we demonstrate that the new methods work excellently. (C) 2011 Elsevier B.V. All rights reserved.

We present the magnetic phase diagram of the nu=2 quantum Hall system on the whole (r(s),E(Z)) plane. We fix the phase boundaries of the paramagnetic and ferromagnetic states by looking for a softening of spin-density excitations in the time-dependent Hartree-Fock theory. A nontrivial phase is obtained in the self-consistent Hartree-Fock theory for r(s)similar to 2 and E(Z)less than or similar to 0.06h omega(c), where both the paramagnetic and ferromagnetic states show spin instability. We show that the obtained phase is the spin-density wave (SDW) state, and explain the mechanism how the SDW stabilizes.

We present the magnetic phase diagram on the (r(s), E(z)) plane in the self-consistent Hartree-Fock theory for the nu = 2 quantum Hall system. We show that there is an area on the (r(s), E(z)) plane where a spiral spin density wave state is the ground state of the system.

We revisit the Bogoliubov transformation as a representation of the group of unitary operators of Balian and Brezin. We show that the group property is best utilized when we treat successive transformations of quasiparticles and their vacuum at the same time. In particular, we establish a one-to-one correspondence between sets of quasiparticle operators and their vacua using the group property. The correspondence determines the quasiparticle vacuum uniquely including the phase, which is inevitable in treating probability amplitudes in a consistent fashion. (C) 2008 Elsevier B.V. All rights reserved.

We present the biorthogonal complement of the standing wave state and its relation to scattering operators. This establishes a completeness relation in terms of the standing wave states, and therefore completes the R-matrix (reactance matrix) theory of scattering. We clarify with an example that the R-matrix theory, in its formulation presented here, can claim its due place in addition to the standard T-matrix theory.

We investigate possible spin instabilities of integer quantum Hall systems as a function of the density within the framework of the self-consistent Hartree-Fock theory. We have found a spin density wave (SDW) state that is lower in energy than the paramagnetic and ferromagnetic states for the filling factor nu=2 in the density region 2.01<r(s)<2.15. The new SDW phase is the very one that has been expected by the softening of excitation spectra of the spin density excitation. We have examined the SDW phase in detail using the density matrix.

Multiple scattering processes in two-dimensional electron systems with an arbitrary spin polarization are expressed as a spin-dependent effective interaction operator, which allows applications in various two-dimensional electron systems. Effects of the spin polarization on the correlation energy and the pair correlation function are discussed in detail in connection with the polarization-dependence of the effective interaction.

A fully microscopic derivation is proposed for an effective interaction operator between electrons in the two-dimensional electron gas (2DEG), which represents multiple-scattering processes in the medium. The obtained interaction features short-range behaviors between electrons, and is presented in a simple form which allows applications in various systems. Short-range correlation in the 2DEG is discussed in detail in terms of the effective interaction with special emphasis on the nonlocal aspect of the correlation. © 2004 The American Physical Society.

We study the spin density response function in a two-dimensional electron system under a strong perpendicular magnetic field. The theoretical framework is the extended RPA which takes into account all the second order self-energies of particle-hole propagators. Our results reproduce the data obtained by a light scattering experiment. We clarify how the extended RPA explains the empirical findings which are different even qualitatively from the RPA response. Especially the important role played by the two-particle-two-hole degrees of freedom is stressed.

We propose an extended Lipkin model that can describe decay processes of particle-hole states and is still solvable. We examine several RPA-type theories using the model. We show explicitly the roles played by the self-energies of particle-hole propagators, and clarify how the extended RPA theory describes collective states better than other RPA-type theories.

We consider the regularized approximations of the discontinuity-inducing singular zero-range interaction in one-dimensional quantum mechanics. We prove that two known approximations, one in terms of local potential and the other in terms of separable function are equivalent. We further show that the interaction allows the perturbative treatment through the coupling constant renormalization.

Ground-state properties of the quasi-one-dimensional electron gas in a quantum wire are calculated in the random-phase approximation (RPA), the ladder approximation, and the Singwi-Tosi-Land-Sjolander approximation. Numerical results are given for the exchange-correlation energy and the compressibility as a function of the electron density and the width of the wire. The dielectric response of the system has been calculated in the local field approximation and compared with the RPA result.

We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G(q), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.

We show explicitly that a negative value of the static dielectric function necessarily means the existence of what might be called a ''ghost plasmon,'' a pole of the screened response chi(SC)(q, omega) on the imaginary axis of omega. The ghost plasmon is a collective state exhausting most of the conductivity sum rule. This is demonstrated for a two-dimensional electron gas with a microscopic calculation.

We show explicitly that a negative value of the static dielectric function necessarily means the existence of what might be called a “ghost plasmon,” a pole of the screened response (Formula presented) on the imaginary axis of ω. The ghost plasmon is a collective state exhausting most of the conductivity sum rule. This is demonstrated for a two-dimensional electron gas with a microscopic calculation. © 1997 The American Physical Society.

The Dyson equation for the particle-hole Green's function, including exchange matrix elements, has been solved exactly for the effective interaction between two electrons in a two-dimensional electron gas. The effective interaction is obtained numerically by solving the Bethe-Goldstone integral equation in a two-dimensional electron gas. The effect of short-range correlations on static and dynamic dielectric functions is studied. Results are compared with the normal random-phase approximation, local-field theories, and recent quantum Monte Carlo results.

The Dyson equation for the particle-hole Green’s function, including exchange matrix elements, has been solved exactly for the effective interaction between two electrons in a two-dimensional electron gas. The effective interaction is obtained numerically by solving the Bethe-Goldstone integral equation in a two-dimensional electron gas. The effect of short-range correlations on static and dynamic dielectric functions is studied. Results are compared with the normal random-phase approximation, local-field theories, and recent quantum Monte Carlo results. © 1996 The American Physical Society.

The Dyson equation for the particle-hole Green's function, including Coulomb exchange matrix elements, has been solved exactly for a two-dimensional electron gas. Static and dynamic dielectric functions have been calculated and compared with normal random-phase-approximation and recent quantum Monte Carlo results.

The effects of short-range electronic correlations on the properties of sodium clusters are studied using the Brueckner g matrix as an effective interaction which describes the scattering of two electrons in the presence of a many-electron medium. The associated cluster Hamiltonian is solved within the Hartree-Fock approximation for the ground state and the dipole plasmon resonance is studied using the self-consistent random-phase approximation. Effects due to ionic core electrons are considered within the pseudojellium model of metal cluster, which goes beyond jellium by using ionic pseudo-Hamiltonians.

We investigate the role of electron correlations on the plasmon dispersion in alkali metals using a sum rule approach. The single pair contribution to the low-energy part of the excitation spectrum is calculated in the framework of the Landau theory of Fermi liquids and used to estimate the plasmon contribution to the compressibility sum rule. The plasmon contribution to the f-sum rule is calculated employing a non-local effective interaction (g-matrix) recently proposed in the literature. The analysis accounts for the strong density dependence of the dispersion coefficient alpha exhibited by experimental data. The average energy of multipair excitations is also estimated.

A relation is studied between the momentum distribution and the response function of nuclei in the quasielastic region. It is shown that a consistent treatment of the ground-state correlation leads simultaneously to (i) the existence of the high-momentum tail of the momentum distribution and to (ii) a quenching of the response function around the quasielastic peak region and an enhancement of the high-energy tail part, i.e. they are two different manifestations of the ground-state correlation.

The charge longitudinal response function is examined in the framework of the random-phase approximation in an isospin-asymmetric nuclear matter where proton and neutron densities are different. This asymmetry changes the response through both the particle-hole interaction and the free particle-hole polarization propagator. We discuss these two effects on the response function on the basis of our numerical results in detail.

We study the isospin-dependent component of the effective nucleon-nucleon interaction which causes the DELTA-T = 1 (p,p') and (p,n) reactions off nuclei. It is shown that, at intermediate energies, the modification to the impulse approximation comes from the g-matrix-type correction and the rearrangement term. They are numerically estimated with the isospin-asymmetric nuclear-matter reaction matrix approach. The isobaric-analog transitions Ca-42(p,n)Sc-44 and 48Ca(p,n)Sc-48 are analyzed.

Second-order perturbation theory with a G-matrix is adopted to examine from a unified point of view the effects of two-particle-two-hole correlations on the matter and momentum distributions and the occupation numbers in Ca-40 and Ca-48. Polarization effects induced by the neutron excess in Ca-48 are investigated in detail.

The norm of the ground-state vector of a many-fermion system that is generated by the adiabatic theorem is examined in detail. A systematic way is presented to renormalize the state vector so that its norm would be conserved during the adiabatic development.

Two expressions for the non-energy-weighted sum rules are studied in detail. One of them contains the information of one- and two-body density matrices of the ground state of a nucleus. The other summarizes the elastic and inelastic processes which determine the nuclear response to an external probe. These processes are well described by the response function method. In this paper, we establish the relation of these two expressions; namely, we show which processes of the response have the information of the one- and two-body density matrices. To make the physical points clear, we use the Coulomb sum rule as an example.

We investigate in detail the Coulomb sum rule of Ca-40 with momentum transfer q = 2.0 fm-1, in the framework of traditional nuclear many-body theory, i.e. by using the G-matrix and the extended RPA theory. We have separately evaluated the one-body part (total charge Z) and the two-body part (two-body correlation function) of the sum rule. A considerable part of the missing charge in the existing experimental data is shown to be explained by the ground-state correlations induced by the tensor force. It is also found that the one-body part is not fully observed in the existing experimental data. Problems left to be discussed are pointed out.