The 4 He3 bound states and the scattering of a 4 He atom off a 4 He dimer at ultra-low energies are investigated using a hard-core version of the Faddeev differential equations. Various realistic 4 He-4 He interactions were employed, amomg them the LM2M2 potential by Aziz and Slaman and the recent TTY potential by Tang, Toennies and Yiu. The ground state and the excited (Efimov) state obtained are compared with other results. The scattering lengths and the atom-diatom phase shifts were calculated for center of mass energies up to 2.45 mK. It was found that the LM2M2 and TTY potentials, although of quite different structure, give practically the same bound-state and scattering results.

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 4 He 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.LANL E-print physics/9612012.

Abstract. We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.

We introduce a new concept of unbounded solutions to the operator Riccati equation A 1 X − XA 0 − XV X + V * = 0 and give a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix. We also provide a new characterization of the set of all contractive solutions under the assumption that the Riccati equation has a contractive solution associated with a spectral subspace of the operator B. In this case we establish a criterion for the uniqueness of contractive solutions.

We consider operator matrices H = ( Blo '01 Al ) with self-adjoint entries A ; , i = 0,1, ead bounded Bol = Bio, acting in the orthogonal rum 31 = 36 @ 311 of Hilbert spacea % and 311. We are especially interested in the eyc where the rpsctrum of, w, A1 ir partly or totally embedded into the continuous spectrum of A0 m d the t r d e r fuoction M~( z )admits mdytic continuation (M an o p e r a t o r -d u d function) through the cuts dong branchea of the continuous spectrum of the entry & into the unphysical wheet(8) of the spectral parameter plane. The d u a s of L in the unphylicrl beet# whom MT1(z) and coneequently the m l v e n t (H -I)-' have polas are w u d y u l l d raonums. A main god of the present work is to 6nd non-selfadjoint operators whore rpactra include the re8onanm an 4 IB to study the completeneea and basis properties of the rasonmce eigenvactars of M l ( z ) in 311. 'It, this end we first construct an operator-valued function R ( Y ) on the #pace of operators in 311 possessing the property: Vl(Y)+l = V~(Z)$I for m y eigenvector $1 of Y corrasponding to an eigendue z and then study the quation H1 = A1 + R(H1). We prove the d n b i l i t y of this quation even in the c a~e where the spectra of A0 and A1 overlap. Uiing the frct that the root vectors of the wlutiom H1 are at the uune time such vectors for M1 (z), we prove completeneon and even b i s propertiea for the root vectors (including those for the resonmces). 1991 M a h m o t i c r Subject Clorsifiation. Primary: 47A56,47Nxx; Secondary: 47N5OI47A40. for any eigenvector $('I corresponding to an eigenvalue z of the operator K. The desired operator Hi was searched for as a solution of the operator equation ( 1-81 Hi = A, + K ( H i ) , i = 0, 1. Notice that an equation of the form (1.8) first appeared explicitly in the paper [7] by M. A. BRAUN. Obviously, if Hi is a solution of Eq. (1.8) and Hi$(i) = z$J(') then, due to (1.7), automatically a$(') = (A< + &(Hi))$(i) = (Ai + &(z))$(') and, thus, for these z and $('I the equality (1.5) holds. The solvability of the equation (1.8) was announced in [29] and proved in [28, 301 under the assumption wketfi %r the Hifbert-Schmidt norm of the couplings Bi,. It was found (28, 301 that the problem of constructing the operators Hi is closely related to the problem of searching for the invariant subspaces 9il i = 0,1, of the matrix H which admit the graph representations A = { u E H : u = ( * , U , O , , ) , U o E 3 1 0 } 9 91 = ( u E 3 1 : u = ( Q O :~) , u 1 € H 1 } , (1.10) strictly below the spectrum of the other one, say (1.15) maxo(A1) < mina(A0). Soon, the result of [2] was extended by V. M. ADAMYAN, H. LANGER, R. MENNICKEN and J . SAURER [3] to the case where (1.16) maxa(A1) 5 mino(A0) and where the couplings Bij were allowed to be unbounded operators such that, for < mino(&), the product (& -a~)-'/~Bol makes sense as a bounded operator. The conditions (1.15), (1.16) were then somewhat weakened by R. MENNICKEN and A. A. SHKALIKOV [27] in the case of a bounded entry A1 and the same type of entries Bij a...