The inverse of an $infty times infty$ symmetric band matrix can be constructed in terms of a matrix continued fraction. For Hamiltonians with Coulomb plus polynomial potentials, this results in an exact and analytic Greens operator which, even in fin
ite-dimensional representation, exhibits the exact spectrum. In this work we propose a finite dimensional representation for the potential operator such that it retains some information about the whole Hilbert-space representation. The potential should be represented in a larger basis, then the matrix should be inverted, then truncated to the desired size, and finally inverted again. This procedure results in a superb low-rank representation of the potential operator. The method is illustrated with a typical nucleon-nucleon potential.
The fishbone potential of composite particles simulates the Pauli effect by nonlocal terms. We determine the $n-alpha$ and $p-alpha$ fish-bone potential by simultaneously fitting to the experimental phase shifts. We found that with a double Gaussian
parametrization of the local potential can describe the $n-alpha$ and $p-alpha$ phase shifts for all partial waves.
By solving the Faddeev equations we calculate the mass of the strange baryons in the framework of a relativistic constituent quark model. The Goldstone-boson-exchange quark-quark interaction is derived from $SU(3)_F$ symmetry, which is explicitly bro
ken as the strange quark is much heavier. This broken symmetry can nicely be accounted for in the Faddeev framework.
The fishbone potential of composite particles simulates the Pauli effect by nonlocal terms. We determine the $alpha-alpha$ fishbone potential by simultaneously fitting to two-$alpha$ resonance energies, experimental phase shifts and three-$alpha$ bin
ding energies. We found that essentially a simple gaussian can provide a good description of two-$alpha$ and three-$alpha$ experimental data without invoking three-body potentials.
A method is presented that allows to solve the Faddeev integral equations of the semirelativistic constituent quark model. In such a model the quark-quark interaction is modeled by a infinitely rising confining potential and the kinetic energy is tak
en in a relativistic form. We solve the integral equations in Coulomb-Sturmian basis. This basis facilitate an exact treatment of the confining potentials.
A formalism is presented that allows an asymptotically exact solution of non-relativistic and semi-relativistic two-body problems with infinitely rising confining potentials. We consider both linear and quadratic confinement. The additional short-ran
ge terms are expanded in a Coulomb-Sturmian basis. Such kinds of Hamiltonians are frequently used in atomic, nuclear, and particle physics.
Three-body resonances in atomic systems are calculated as complex-energy solutions of Faddeev-type integral equations. The homogeneous Faddeev-Merkuriev integral equations are solved by approximating the potential terms in a Coulomb-Sturmian basis. T
he Coulomb-Sturmian matrix elements of the three-body Coulomb Greens operator has been calculated as a contour integral of two-body Coulomb Greens matrices. This approximation casts the integral equation into a matrix equation and the complex energies are located as the complex zeros of the Fredholm determinant. We calculated resonances of the e-Ps system at higher energies and for total angular momentum L=1 with natural and unnatural parity
Two body data alone cannot determine the potential uniquely, one needs three-body data as well. A method is presented here which simultaneously fits local or nonlocal potentials to two-body and three-body observables. The interaction of composite par
ticles, due to the Pauli effect and the indistinguishability of the constituent particles, is genuinely nonlocal. As an example, we use a Pauli-correct nonlocal fish-bone type optical model for the $alpha-alpha$ potential and derive the fitting parameters such that it reproduces the two-$alpha$ and three-$alpha$ experimental data.
