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We consider the family of $3 \times 3$ operator matrices ${\bf H}(K),$ $K \in {\Bbb T}^3:=(-\pi; \pi]^3$ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set $\Lambda \subset {\Bbb T}^3$ to prove the existence of infinitely many eigenvalues of ${\bf H}(K)$ for all $K \in \Lambda$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K \in \Lambda,$ the number $N(K, z)$ of eigenvalues of ${\bf H}(K)$ lying on the left of $z,$ $z<0,$ satisfies the asymptotic relation $\lim\limits_{z \to -0} N(K, z) |\log|z||^{-1}={\mathcal U}_0$ with $0<{\mathcal U}_0<\infty,$ independently on the cardinality of $\Lambda.$ Moreover, we prove that for any $K \in \Lambda$ the operator ${\bf H}(K)$ has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model., Comment: 18 pages

We consider the collinear ternary fission which is a sequential ternary decay with a very short time between the ruptures of two necks connecting the middle cluster of the ternary nuclear system and outer fragments. In particular, we consider the case where the Coulomb field of the first massive fragment separated during the first step of the fission produces a lower pre-scission barrier in the second step of the residual part of the ternary system. In this case, we obtain a probability of about $10^{-3}$ for the yield of massive clusters such as \nuclide[70]{Ni}, \nuclide[80-82]{Ge}, \nuclide[86]{Se}, and \nuclide[94]{Kr} in the ternary fission of \nuclide[252]{Cf}. These products appear together with the clusters having mass numbers of $A = 132$--$140$. The results show that the yield of a heavy cluster such as \nuclide[68-70]{Ni} would be followed by a product of $A = 138$--$148$ with a large probability as observed in the experimental data obtained with the FOBOS spectrometer at the Joint Institute for Nuclear Research. The third product is not observed. The landscape of the potential energy surface shows that the configuration of the Ni + Ca + Sn decay channel is lower about 12 MeV than that of the Ca + Ni + Sn channel. This leads to the fact, that the yield of Ni and Sn is large. The analysis on the dependence of the velocity of the middle fragment on mass numbers of the outer products leads to the conclusion that, in the collinear tripartition channel of \nuclide[252]{Cf}, the middle cluster has a very small velocity, which does not allow it to be found in experiments., Comment: 11 pages, 9 figures

The existence of a new type of cluster decay called "collinear cluster tri-partition" (CCT) is discussed by an analysis of the landscape of the potential energy surface (PES). The total energy of the ternary system is found as a sum the binding energies of fragments and nucleus-nucleus interaction between them. The pre-scission state of the ternary system is assumed to be arranged as a chain of the three fragments along a straight line. Minima and valleys of the PES are determined by variation of the proton and neutron distributions between them. Pre-scission prompt emission of neutrons is assumed and PES is calculated for the cases of emission of 2---4 neutrons. The presence of the valley corresponding to the formation of the isotopes of Sn with masses $A$=130---136 is inherent for all PES calculated for CCT for spontaneous fission of $^{252}$Cf and fission induced by neutrons of $^{235}$U. There are local minima indicating the formation of Ca, Fe, Ni, Ge and Se isotopes having magic proton or/and neutron numbers, such as 20, 28, and 50. The analysis shows that the experimentally observed $^{68}$Ni is formed as the edge fragment of the ternary system connecting to Sn by Si and Ca isotopes at fission of $^{236}$U and $^{252}$Cf, respectively., Comment: 13 pages, 6 figures

The fusion and evaporation residue cross sections for the $^{50}$Ti+$^{249}$Cf and $^{54}$Cr+$^{248}$Cm reactions calculated by the combined dinuclear system and advanced statistical models are compared. These reactions are considered to be used to synthesize the heaviest superheavy element. The $^{50}$Ti+$^{249}$Cf reaction is more mass asymmetric than $^{54}$Cr+$^{248}$Cm and the fusion excitation function for the former reaction is higher than the one for the latter reaction. The evaporation residue excitation functions for the mass asymmetric reaction is higher in comparison with the one of the $^{54}$Cr+$^{248}$Cm reaction. The use of the mass values of superheavy nuclei calculated in the framework of the macroscopic-microscopic model by the Warsaw group leads to smaller evaporation residue cross section for both the reactions in comparison with the case of using the masses calculated by Peter M\"oller {\it et al}. The $^{50}$Ti+$^{249}$Cf reaction is more favorable in comparison with the $^{54}$Cr+$^{248}$Cm reaction: the maximum values of the excitation function of the 3n-channel of the evaporation residue formation for the $^{50}$Ti+$^{249}$Cf and $^{54}$Cr+$^{248}$Cm reactions are about 0.1 and 0.07 pb, respectively, but the yield of the 4n-channel for the former reaction is lower (0.004 pb) in comparison with the one (0.01 pb) for the latter reaction., Comment: 21 pages, 10 figures, 2 tables

In the present paper our aim is to explore some spectral properties of the family two-particle discrete Schr\"odinger operators $h^{\mathrm{d}}(k)=h^{\mathrm{d}}_0(k)+ \mathbf{v},$ $k\in \T^\mathrm{d},$ on the $\mathrm{d}$ dimensional lattice $\Z^{\mathrm{d}},$ $\mathrm{d}\geq 1,$ $k$ being the two-particle quasi-momentum. Under some condition in the case $k\in \T^{\mathrm{d}}\setminus (-\pi,\pi)^{\mathrm{d}},$ we establish necessary and sufficient conditions for existence of infinite discrete spectrum of the operator $h^{\mathrm{d}}(k)$., Comment: 11 pages

A Hamiltonian (model operator) $H$ associated to a quantum system describing three particles in interaction, without conservation of the number of particles, is considered. The Faddeev type system of equations for eigenvectors of $H$ is constructed. The essential spectrum of $H$ is described by the spectrum of the channel operator.

The mixing of the quasifission component to the fissionlike cross section causes ambiguity in the quantitative estimation of the complete fusion cross section from the observed angular and mass distributions of the binary products. We show that the partial cross section of quasifission component of binary fragments covers the whole range of the angular momentum values leading to capture. The calculated angular momentum distributions for the compound nucleus and dinuclear system going to quasifission may overlap: competition between complete fusion and quasifission takes place at all values of initial orbital angular momentum. Quasifission components formed at large angular momentum of the dinuclear system can show isotropic angular distribution and their mass distribution can be in mass symmetric region similar to the characteristics of fusion-fission components. As result the unintentional inclusion of the quasifission contribution into the fusion-fission fragment yields can lead to overestimation of the probability of the compound nucleus formation., Comment: 15 pages, 6 figures, International Conference on Nuclear Reactions on Nucleons and Nuclei, Messina, Italy, October 5-9, 2009

The excitation functions of capture, complete fusion, and evaporation residue formation in the $^{16}$O+$^{184}$W and $^{19}$F+$^{181}$Ta reactions leading to the same $^{200}$Pb compound nucleus has been studied theoretically to explain the experimental data showing more intense yield of evaporation residue in the former reaction in comparison with that in the latter reaction. The observed difference is explained by large capture cross section in the former and by increase of the quasifission contribution to the yield of fission-like fragments in the $^{19}$F+$^{181}$Ta reaction at large excitation energies. The probability of compound nucleus formation in the $^{16}$O+$^{184}$W reaction is larger but compound nuclei formed in both reactions have similar angular momentum ranges at the same excitation energy. The observed decrease of evaporation residue cross section normalized to the fusion cross section in the $^{19}$F+$^{181}$Ta reaction in comparison with the one in the $^{16}$O+$^{184}$W reaction at high excitation energies is explained by the increase of hindrance in the formation of compound nucleus connected with more quick increase of the quasifission contribution in the $^{19}$F induced reaction. The spin distributions of the evaporation residue cross sections for the two reactions are also presented., Comment: 11 pages, 5 figures

The yields of evaporation residues, fusion-fission and quasifission fragments in the $^{48}$Ca+$^{144,154}$Sm and $^{16}$O+$^{186}$W reactions are analyzed in the framework of the combined theoretical method based on the dinuclear system concept and advanced statistical model. The measured yields of evaporation residues for the $^{48}$Ca+$^{154}$Sm reaction can be well reproduced. The measured yields of fission fragments are decomposed into contributions coming from fusion-fission, quasifission, and fast-fission. The decrease in the measured yield of quasifission fragments in $^{48}$Ca+$^{154}$Sm at the large collision energies and the lack of quasifission fragments in the $^{48}$Ca+$^{144}$Sm reaction are explained by the overlap in mass-angle distributions of the quasifission and fusion-fission fragments. The investigation of the optimal conditions for the synthesis of the new element $Z$=120 ($A$=302) show that the $^{54}$Cr+$^{248}$Cm reaction is preferable in comparison with the $^{58}$Fe+$^{244}$Pu and $^{64}$Ni+$^{238}$U reactions because the excitation function of the evaporation residues of the former reaction is some orders of magnitude larger than that for the last two reactions., Comment: 27 pages, 12 figures, submitted to Phys. Rev. C

The influence of the orientation angles of the target nucleus symmetry axis relative to the beam direction on the production of the evaporation residues is investigated for the $^{48}$Ca+$^{154}$Sm reaction as a function of the beam energy. At low energies ($E_{\rm c.m.}<$137 MeV), the yield of evaporation residues is observed only for collisions with small orientation angles ($\alpha_T<45^0$). At large energies (about $E_{\rm c.m.}=$140--180 MeV) all the orientation angles $\alpha_T$ can contribute to the evaporation residue cross section $\sigma_{ER}$ in the 10--100 mb range, and at $E_{c.m.}>$180 MeV $\sigma_{ER}$ ranges around 0.1--10 mb because the fission barrier for a compound nucleus decreases by increasing its excitation energy and angular momentum., Comment: 20 pages, 10 figures, submitted to JPSJ

A model operator $H$ associated to a system describing four particles in interaction, without conservation of the number of particles, is considered. We describe the essential spectrum of $H$ by the spectrum of the channel operators and prove the Hunziker-van Winter-Zhislin (HWZ) theorem for the operator $H.$ We also give some variational principles for boundaries of the essential spectrum and interior eigenvalues.

The anisotropy in the angular distribution of the fusion-fission and quasifission fragments for the $^{16}$O+$^{238}$U, $^{19}$F+$^{208}$Pb and $^{32}$S+$^{208}$Pb reactions is studied by analyzing the angular momentum distributions of the dinuclear system and compound nucleus which are formed after capture and complete fusion, respectively. The orientation angles of axial symmetry axes of colliding nuclei to the beam direction are taken into account for the calculation of the variance of the projection of the total spin onto the fission axis. It is shown that the deviation of the experimental angular anisotropy from the statistical model picture is connected with the contribution of the quasifission fragments which is dominant in the $^{32}$S+$^{208}$Pb reaction. Enhancement of anisotropy at low energies in the $^{16}$O+$^{238}$U reaction is connected with quasifission of the dinuclear system having low temperature and effective moment of inertia., Comment: 17 pages 8 figures. Submitted to Euro. Phys. Jour. A

A family of Friedrichs models under rank one perturbations $h_{\mu}(p),$ $p \in (-\pi,\pi]^3$, $\mu>0,$ associated to a system of two particles on the three dimensional lattice $\Z^3$ is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of $h_\mu(p)$ for all nontrivial values of $p$ under the assumption that $h_\mu(0)$ has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained., Comment: 15 pages

A model operator $H$ associated to a system of three-particles on the three dimensional lattice $\Z^3$ and interacting via pair non-local potentials is studied. The following results are proven: (i) the operator $H$ has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point, in the case, where both Friedrichs model operators $h_{\mu_\alpha}(0),\alpha=1,2,$ have threshold resonances. (ii) the operator $H$ has a finite number of eigenvalues lying outside of the essential spectrum, in the case, where at least one of $h_{\mu_\alpha}(0), \alpha=1,2,$ has a threshold eigenvalue., Comment: 14 pages

For a wide class of two-body energy operators $h(k)$ on the three-dimensional lattice $\bbZ^3$, $k$ being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values $k$, $k\ne 0$, the discrete spectrum of $h(k)$ below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian $h(0)$ corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

A system of three quantum particles on the three-dimensional lattice $\Z^3$ with arbitrary "dispersion functions" having non-compact support and interacting via short-range pair potentials is considered. The energy operators of the systems of the two-and three-particles on the lattice $\Z^3$ in the coordinate and momentum representations are described as bounded self-adjoint operators on the corresponding Hilbert spaces. For all sufficiently small nonzero values of the two-particle quasi-momentum $k\in (-\pi,\pi]^3$ the finiteness of the number of eigenvalues of the two-particle discrete Schr\"odinger operator $h_\alpha(k)$ below the continuous spectrum is established. A location of the essential spectrum of the three-particle discrete Schr\"odinger operator $H(K),K\in (-\pi,\pi]^3$ the three-particle quasi-momentum, by means of the spectrum of $h_\alpha(k)$ is described. It is established that the essential spectrum of $H(K), K\in (-\pi,\pi]^3$ consists of a finitely many bounded closed intervals.

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice $\Z^3$ and interacting via zero-range attractive potentials is considered. For the two-particle energy operator $h(k),$ with $k\in \T^3=(-\pi,\pi]^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of $h(k)$ for $k\neq0$ is proven, provided that $h(0)$ has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schr\"{o}dinger operator $H(K), K\in \T^3$ being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number $N(0,z)$ of eigenvalues of H(0) lying below $z<0$ the following limit exists $$ \lim_{z\to 0-} \frac {N(0,z)}{\mid \log\mid z\mid\mid}=\cU_0 $$ with $\cU_0>0$. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum $K$ the finiteness of the number $ N(K,\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the essential spectrum is established and the asymptotics for the number $N(K,0)$ of eigenvalues lying below zero is given., Comment: 28 pages

The effects of the entrance channel and shell structure of reacting nuclei on the experimental evaporation residues have been studied by analysing the 40Ar+176Hf, 86Kr+130,136Xe, 124Sn+92Zr and 48Ca+174Yb reactions leading to the 216Th* and 222Th* compound nuclei. The measured excitation function of evaporation residues for the 124Sn+92Zr reaction was larger than that for the 86Kr+130Xe reaction. The experimental values of evaporation residues in the 86Kr+136Xe reaction were about 500 times larger than that in the 86Kr+130Xe reaction. These results are explained by the initial angular momentum dependence of the fusion excitation functions calculated in framework of the dinuclear system concept and by the differences in survival probabilities calculated in framework of advanced statistical model. The dependencies of the fission barrier and the \Gamma_n / \Gamma_f ratio on the angular momentum of the excited compound nucleus are taken into account., Comment: 13 pages, 9 (eps) figures, uses revtex4, submitted to Phys.Rev. C

The role of the entrance channel in fusion-fission reactions was studied by the theoretical analysis of the experimental evaporation residue excitation functions for reactions leading to the same compound nucleus. The evaporation residues cross sections for xn-channels were calculated in the frame of the combined dinuclear system concept and advanced statistical model. The revealed differences between experimental data on the evaporation residues in the ^{40}Ar+^{176}Hf, ^{86}Kr + ^{130}Xe and ^{124}Sn + ^{92}Zr reactions leading to the ^{216}Th^* compound nucleus are explained by the different spin distributions of compound nuclei which are formed. It is shown that the intrinsic fusion barrier B^*_{fus} and size of potential well are different for every entrance channel., Comment: 14 pages, 5 EPS figures, Latex, ws-p8-50x6-00.cls is used, Talk given at the Conference NPBL2001, Lipari (Messina, Italy), 21-24 May, 2001

The role of the entrance channel in the fusion-fission reactions leading to nearly the same superheavy compound nucleus is studied in the framework of dynamic model. The calculations are done for $^{48}$Ca + $^{244}$Pu and $^{74,76}$Ge + $^{208}$Pb reactions which could lead to formation of superheavy element Z=114. It is shown that for these reactions there is an energy window for the values of the bombarding energy at which a capture probability is sufficiently large. Together with the restriction coming from the intrinsic barrier for fusion, it helps to find an optimal value of the bombarding energy for a given projectile-target combination., Comment: 7 pages, Latex, 6 Postscript figures, replaced due to the minor mistake in Latex command

Based on the microscopic model, the friction coefficient for the relative motion of nuclei in deep-inelastic heavy-ion collisions is calculated. The radial dependence of the friction coefficient is studied and the results are compared with those found by other methods. Based on this result, it was demonstrated that the kinetic energy dissipation in deep-inelastic heavy-ion collisions is a gradual process which takes up a significant part of a reaction time. An advantage of the suggested method is that it allows one to consider the relative motion of nuclei and the intrinsic motion self-consistently., Comment: 15 pages, RevTex, 7 Postscript figures, submitted to Phys. Rev. C