arXiv subjects
Publications and source records attributed to A. K. Motovilov.
Algorithm, based on explicit representations for the analytic continuation of Faddeev components of the three-body T-matrix in unphysical energy sheets, is employed to study mechanism of disappearance and formation of the Efimov levels of the helium ^4He trimer.
The spectral problem (A + V(z))ψ=zψis considered with A, a self-adjoint operator. The perturbation V(z) is assumed to depend on the spectral parameter z as resolvent of another self-adjoint operator A': V(z)=-B(A'-z)^{-1}B^{*}. It is supposed that the operator B has a finite Hilbert-Schmidt norm and spectra of the operators A and A' are separated. Conditions are formulated when the perturbation V(z) may be replaced with a ``potential'' W independent of z and such that the operator H=A+W has the same spectrum and the same eigenfunctions (more precisely, a part of spectrum and a respective part of eigenfunctions system) as the initial spectral problem. The operator H is constructed as a solution of the non-linear operator equation H=A+V(H) with a specially chosen operator-valued function V(H). In the case if the initial spectral problem corresponds to a two-channel variant of the Friedrichs model, a basis property of the eigenfunction system of the operator H is proved. A scattering theory is developed for H in the case where the operator A has continuous spectrum.
A mechanism of disappearance and formation of the Efimov levels of the helium ^4He_3 trimer is studied when the force of interatomic interaction is changed. The resonances including virtual levels are calculated by the method based on the solving the boundary value problem, at complex energies, for the Faddeev differential equations describing the (2+1 --> 2+1; 1+1+1) scattering processes.
We present a mathematically rigorous method suitable for solving three-body bound state and scattering problems when the inter-particle interaction is of a hard-core nature. The proposed method is a variant of the Boundary Condition Model and it has been employed to calculate the binding energies for a system consisting of three ^4He atoms. Two realistic He-He interactions of Aziz and collaborators, have been used for this purpose. The results obtained compare favorably with those previously obtained by other methods. We further used the model to calculate, for the first time, the ultra-low energy scattering phase shifts. This study revealed that our method is ideally suited for three-body molecular calculations where the practically hard-core of the inter-atomic potential gives rise to strong numerical inaccuracies that make calculations for these molecules cumbersome.
We consider the analytic continuation of the transfer function for a 2x2 matrix Hamiltonian into the unphysical sheets of the energy Riemann surface. We construct a family of non-selfadjoint operators which reproduce certain parts of the transfer-function spectrum including resonances situated on the unphysical sheets neighboring the physical sheet. On this basis, completeness and basis properties for the root vectors of the transfer function (including those for the resonances) are proved.
A mechanism of disappearance and formation of the Efimov levels of the helium ^4He trimer is studied when the force of the interatomic interaction is changed. It is shown that these levels arise from virtual levels which are in turn formed from (quasi)resonances settled on the real axis. The resonances including virtual levels are calculated by the method based on the solution of the boundary value problem, at complex energies, for the Faddeev differential equations describing the scattering processes (2+1\to 2+1; 1+1+1). All the calculations are performed with the known interatomic Aziz He-He potential HFD-B. A very strong repulsive component of this potential at short distances between helium atoms is approximated by a hard core. A special attention is paid to the substantiation of the method used for computing resonances and to the investigation of its applicability range.
Algorithm, based on explicit representations for analytic continuation of the T-matrix Faddeev components on unphysical sheets, is worked out for calculations of resonances in the three-body quantum problem. According to the representations, poles of T-matrix, scattering matrix and Green function on unphysical sheets, interpreted as resonances, coincide with those complex energy values where appropriate truncations of the scattering matrix have zero as eigenvalue. Scattering amplitudes on the physical sheet, necessary to construct scattering matrix, are calculated on the basis of the Faddeev differential equations. The algorithm developed is applied to search for the resonances in the $nnp$ system and in a model three-boson system.
Explicit representations for the Faddeev components of the three-body T-matrix continued analytically into unphysical sheets of the energy Riemann surface are formulated. According to the representations, the T-matrix in unphysical sheets is explicitly expressed in terms of its components taken in the physical sheet only. The representations for the T-matrix are then used to construct similar representations for the analytic continuation of the three-body scattering matrices and the resolvent. Domains on unphysical sheets are described where the representations obtained can be applied. A method for finding three-body resonances based on the Faddeev differential equations is proposed.
We consider operator matrices {\bf H}= (A_0 B_{01} \\ B_{10} A_{1}) with self-adjoint entries A_i, i=0,1, and bounded B_{01}=B_{10}^*, acting in the orthogonal sum {\cal H}={\cal H}_0\oplus{\cal H}_1 of Hilbert spaces {\cal H}_0 and {\cal H}_1. We are especially interested in the case where the spectrum of, say, A_1 is partly or totally embedded into the continuous spectrum of A_0 and the transfer function M_1(z)=A_1-z+V_1(z), where V_1(z)=B_{10}(z-A_0)^{-1}B_{01}, admits analytic continuation (as an operator-valued function) through the cuts along branches of the continuous spectrum of the entry A_0 into the unphysical sheet(s) of the spectral parameter plane. The values of z in the unphysical sheets where M_1^{-1}(z) and consequently the resolvent (H-z)^{-1} have poles are usually called resonances. A main goal of the present work is to find non-selfadjoint operators whose spectra include the resonances as well as to study the completeness and basis properties of the resonance eigenvectors of M_1(z) in {\cal H}_1. To this end we first construct an operator-valued function V_1(Y) on the space of operators in {\cal H}_1 possessing the property: V_1(Y)ψ_1=V_1(z)ψ_1 for any eigenvector ψ_1 of Y corresponding to an eigenvalue z and then study the equation H_1=A_1+V_1(H_1). We prove the solvability of this equation even in the case where the spectra of A_0 and A_1 overlap. Using the fact that the root vectors of the solutions H_1 are at the same time such vectors for M_1(z), we prove completeness and even basis properties for the root vectors (including those for the resonances).
We present a new, mathematically rigorous, method suitable for bound state and scattering processes calculations for various three atomic or molecular systems where the underlying forces are of a hard-core nature. We employed this method to calculate the binding energies and the ultra-low energy scattering phase shifts below as well as above the break-up threshold for the three He-atom system. The method is proved to be highly successful and suitable for solving the three-body bound state and scattering problem in configuration space and thus it paves the way to study various three-atomic systems, and to calculate important quantities such as the cross-sections, recombination rates etc.
We present a mathematically rigorous method for solving three-atomic bound state and scattering problems. The method is well suited for applications in systems where the inter-atomic interaction is of a hard-core nature. It has been employed to obtain the ground- and excited-state energies for the Helium trimer and to calculate, for the first time, the scattering phase shifts and wave-functions for the He atom-He dimer at ultra-low energies.
Algorithm, based on explicit representations for analytic continuation of the T-matrix Faddeev components on unphysical sheets, is worked out for calculations of resonances in the three-body quantum problem. According to the representations, poles of the T--matrix, scattering matrix and resolvent on unphysical sheets, interpreted as resonances, coincide with those complex energy values where appropriate truncations of the scattering matrix have zero as eigenvalue. Scattering amplitudes on the physical sheet, necessary to construct scattering matrix, are calculated on the basis of the Faddeev differential equations. Effectiveness of the algorithm developed is demonstrated for example of searching for resonances in the system $nnp$ and in a model three-boson system.
It is widely accepted that in molecular systems the nuclear interaction plays a negligible role, because of the strong Coulomb repulsion of the nuclei at small distances. We are going to show that this is not always true. The existence of an extended nuclear resonance may lead to considerably enhanced nuclear reaction rates in appropriately prepared molecules. Especially we point out that p+p+ ^{16}O, i.e., the constituents of water, can form a ^{18}Ne(1^-) threshold resonance which decays under energy release into ^{17}F and a proton.
A proof is given for the explicit representations which have been formulated in the author's previous work (nucl-th/9505028) for the Faddeev components of three-body T-matrix continued analytically on unphysical sheets of the energy Riemann surface. Also, the analogous representations for analytical continuation of the three-body scattering matrices and resolvent are proved. An algorithm to search for the three-body resonances on the base of the Faddeev differential equations is discussed.
Explicit representations are formulated for the Faddeev components of three-body T-matrix continued analytically on unphysical sheets of the energy Riemann surface. According to the representations, the T-matrix on unphysical sheets is obviously expressed in terms of its components taken on the physical sheet only. The representations for T-matrix are used then to construct similar representations for analytical continuation of three-body scattering matrices and resolvent. Domains on unphysical sheets are described where the representations obtained can be applied.
The probability of nuclear transitions $p+p+{}^{16}{O} \, \rightarrow \, {}^{18}{Ne}\, (4.522,1^-)$ in molecular water is estimated. Due to the practically exact agreement of the energy of the $Ne$ resonance and of the $p+p+{}^{16}{O}$ threshold, the transition probability is found to be considerably enhanced. This indicates the possibility of nuclear fusion in rotationally excited $H_2O$ molecules in the state $1^-$.
Systems of three and four quantum particles in the boundary-condition model are considered. The Faddeev-Yakubovsky approach is applied to construct the Fredholm-type integral equations for these systems in framework of the Potential theory. The boundary-value problems are formulated for the Faddeev-Yakubovsky components of wave functions.
The spectral problem $(A + V(z))ψ=zψ$ is considered where the main Hamiltonian $A$ is a self-adjoint operator of sufficiently arbitrary nature. The perturbation $V(z)=-B(A'-z)^{-1}B^{*}$ depends on the energy $z$ as resolvent of another self-adjoint operator $A'$. The latter is usually interpreted as Hamiltonian describing an internal structure of physical system. The operator $B$ is assumed to have a finite Hilbert-Schmidt norm. The conditions are formulated when one can replace the perturbation $V(z)$ with an energy-independent ``potential'' $W$ such that the Hamiltonian $H=A +W$ has the same spectrum (more exactly a part of spectrum) and the same eigenfunctions as the initial spectral problem. The Hamiltonian $H$ is constructed as a solution of the non-linear operator equation $H=A+V(H)$. It is established that this equation is closely connected with the problem of searching for invariant subspaces of the Hamiltonian $ {\bf H}=\left[ \begin{array}{lr} A & B B^{*} & A' \end{array}\right]. $ The orthogonality and expansion theorems are proved for eigenfunction systems of the Hamiltonian $ H=A + W $. Scattering theory is developed for this Hamiltonian in the case where the operator $A$ has continuous spectrum.