H. van Haeringen

A Characteristic Optimism Factor in Fuzzy Decision-Making

Abstract In multi-objective attribute fuzzy decision making one wishes to select the best alternative. Such a selection is based on many different aspects of varying degrees of importance. To solve the problem of selection one has to find at least approximate values that represent the satisfaction of each alternative to all objectives and also values that represent the relative importance of each objective. Moreover, the operation necessary to combine the values corresponding to the objectives is not uniquely defined and has to be chosen. This paper presents a general formulation of different operators for the computation of these values and provides some insight into the effect of applying a particular operation. This can be considered as a subjective optimism index of the decision maker.

Off‐shell T matrix for Coulomb plus simple separable potentials for all l in closed form

We derive simple exact analytic expressions for the off‐shell T matrices associated with potentials consisting of the sum of the Coulomb potential and any so‐called simple separable potential, for all partial waves. Such potentials play an important role in model calculations for the description of the interaction between charged particles, e.g., protons. We also derive some interesting properties of the Coulomb‐modified form factors associated with the form factors of the separable potentials. The analytic expressions obtained are useful for theoretical and numerical applications.

Coulomb‐modified scattering parameters for Coulomb‐plus‐separable potentials for all l

For three different separable‐potential models, closed analytical expressions are presented for the Coulomb‐modified scattering length and effective range for all values of the angular‐momentum quantum number, l=0,1,2,... . In the derivation of these results, use is made of the regular and irregular Coulomb wave functions that are entire analytic in k2. It is shown that the Coulomb‐modified effective‐range function can be written as a simple expression involving these entire‐analytic functions.

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.

Twenty‐four optimal inequalities and several new representations for the Coulomb T matrices in momentum space

We derive new series and integral representations for the Coulomb transition matrix in momentum space, 〈p‖Tc‖p′〉, and for its partial‐wave projections, 〈p‖Tcl‖p′〉 (l=0,1,...), to be denoted by Tc and Tcl, respectively. We also consider hypergeometric‐function representations for Tc and Tcl and discuss their analytic continuation to the whole complex k plane (k2 is the energy). The new integrals are essentially ∫π0 cosh γt (ρ−cos t)−1 dt for Tc and ∫π0 cosh γt ×Ql(uu′+vv′ cos t)dt for Tcl, where γ is Sommerfeld’s parameter and ρ,u,u′,v, and v′ are variables depending on the energy and the momenta; related integrals follow from these. A well‐known and convenient series representation for Tc consists essentially of the sum ∑nyn(n2+γ2)−1, where y depends on the energy and the momenta. We derive its analog for Tcl, the corresponding sum being ∑n(n2+γ2)−1Qnl(u) P−nl(u′), 1<u′<u. This sum is a new member of the family of sums of products of Legendre functions that can be evaluated in a relatively simple closed f...

Charged-particle off-shell scattering: The T matrix for the pure Coulomb potential and for Coulomb plus simple separable potentials for all partial waves

Abstract We present exact closed analytic expressions for (i) the partial-wave projected off-shell Coulomb T matrix, (ii) the Coulomb Jost state, and (iii) the off-shell T matrix associated with the sum of the Coulomb potential and simple separable potentials, for all partial waves in the momentum representation. These expressions consist of combinations of elementary functions and the extensively studied hypergeometric function2F1(1, iγ; 1+iγ) and are therefore suitable for numerical computations.

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.

Charged‐particle off‐shell scattering: common structure of all two‐body off‐shell scattering quantities for Coulomb plus rational separable potentials

We establish a common structure of all two‐body off‐shell scattering quantities (expressed in momentum space or coordinate space) associated with Coulomb plus rational separable potentials. We present expressions in so‐called maximal‐reduced closed form, including new formulas: (i) for the off‐shell Jost state for the Coulomb potential, (ii) for the off‐shell Jost function associated with the Coulomb plus Yamaguchi potential, and (iii) for the scattering, regular, and Jost states in coordinate representation for Coulomb plus simple separable potentials for all l.

Some infinite series of products of Legendre and gamma functions

We derive closed expressions for some infinite series of products of Legendre functions and gamma functions. A particular series has been used to obtain the partial‐wave projected quantum mechanical Coulomb transition matrix in closed analytic form for all partial waves, l=0, 1,⋅⋅⋅.

A family of sums of products of Legendre functions

We present simple analytic expressions for a few sums of products of Legendre functions, of the type J∞n = 0(2n+1)Pαn(x)Pβn(y) Pγn(z)Qμn(n).