L. P. Kok

On the theory of complex potential scattering

We study the effect of the addition of a complex potential lambdaV/sub sep/ to an arbitrary Schroedinger operator H = H/sub 0/+V on the singularities of the S matrix, as a function of lambda. Here V/sub sep/ is a separable interaction, and lambda is a complex coupling parameter. The paths of these singularities are determined to a great extent by certain saddle points in the momentum (or energy) plane. We explain certain critical phenomena recently reported in the literature. Associated with these saddles are branch-type singularities in the complex lambda plane, which are dynamical in origin. Some examples are discussed in detail.

The Coulomb unitarity relation and some series of products of three Legendre functions

We obtain from the off‐shell Coulomb unitarity relation a closed expression for J∞l = 0(2l+1)Pl(x) ×Qliγ ( y) Ql−iγ (z), and we consider some related series of products of Legendre functions.

Bound-state solution in momentum space of three-particle equations with Coulomb interaction

We solve the bound-state Faddeev equations in the momentum representation for a system of three identical bosons interacting through Yamaguchi forces. Two of the particles are then given an electric charge. The choice of interaction parameters is inspired by the trinucleon systems. In this model we are able to compute the Coulomb energyΔEc of3He without further numerical or analytical approximations by solving the corresponding equations of Veselova and Alt, Sandhas, and Ziegelmann. We then are able to test commonly made approximations: We find that all three possible types of diagrams involving the CoulombT matrixTc contribute substantially. ReplacingTc byVc induces an error of a few percent inΔEc, and simplifies the numerical computations by orders of magnitude. ReplacingVc by itsS-wave projection induces only a small error.

New inequalities for the Coulomb T matrix in momentum space

We present and prove new inequalities for the momentum representation (i) of the full Coulomb transition operator Tc, and (ii) of its partial‐wave projections, Tcl, for all l=0, 1, ... . Also, a previously conjectured inequality is proved: ‖Tc/Vc‖>1, under certain conditions. These inequalities are useful for gaining insight into the—very convenient, and often proposed in the context of three‐particle calculations—approximation in which the Coulomb transition operator is replaced by the Coulomb potential operator. Such an approximation is obviously not accurate at any zero of the T matrix. We investigate the zeros of 〈p‖Tc‖p′〉 for positive energy, and we give simple approximation formulas for these zeros.

Modified effective-range function for two-range potentials

We present the general formula for a modified effective-range function (MERF) which is associated with any potentialV(r) that allows a decomposition into a large-range component and a short-range one. The MERF is a complex real-meromorphic function in the complexk plane in a domain containing the origin. The large-range part of the potential has to satisfy the condition of analyticity atr=0. The extension to the case of the Coulomb and Hulthén potentials which violate this condition is briefly discussed.

Degenerate perturbations in nonrelativistic quantum mechanics

We investigate the effects on the discrete spectrum of an arbitrary quantum mechanical Schrodinger operator H, which are caused by the addition of a real rank N separable potential to H. For such a potential the bound state energies En(λ) as function of the real potential strength λ are in general confined to certain bounded intervals. This remarkable phenomenon can be seen as a particular case of the general situation of complex potential strengths.

ON THE ADEQUACY OF THE UNITARY POLE EXPANSION

Publisher Summary The separability of model nuclear interactions simplifies the three-body problem greatly. Therefore, the separable approximations of local potentials, like the unitary pole expansion (UPE), have been used frequently. In order to judge the quality of such approximations, one usually compares the full T-matrices for the original potential and its approximation. This chapter presents the comparison between the original potential V(r) and the potential V 1 (r), which generates the same on-shell T-matrix and bound state energy as V. It explains the corresponding inverse scattering problem to find the local potential V 1 (r), which is unique. The comparison between V and V 1 is easy because of their dependence on just one variable, whereas the corresponding T-matrices depend on three variables. Unitary pole approximation (UPA) and UPE may be not such good approximations as some three-body calculations seem to indicate. In particular, one should expect deviations at high energies and in many-body problems, the inner part of the inter-action plays a more important role.