Application of the Jost-matrix theory to the lambda-nuclear and multi-lambda systems

Abstract

This study involves the investigation into the hypernuclear and multi-lambda systems using the Jost-function method, as well as the recovery of the two-body potential from the two and three-body systems using the approximate(guessed) wavefunction. The Schrodinger equation describing the quantum system of interest is solved by being replaced with a system of first-order differential equations, which enable one to obtain the Jost functions. These Jost functions are multi-valued energy functions which can be treated as single-valued functions defined on a complex energy surface called the Riemann surface. Direct calculations of the Jost functions, the S-matrix, for all complex momenta of physical interests including the spectral points corresponding to the bound states and resonance states can be obtained. In this work, this method was used to locate the spectral points for the wide range of Λ-nuclear systems within the two-body ΛA-model. The S-matrix residues as well as the corresponding Nuclear-Vertex and Asymptotic-Normalization constants (NVC’s and ANC’s) for the bound states are also found. For scattering parameters the Jost functions were factorized in such a way that they contain certain combination of the channel momenta times an analytic single-valued function of the energy E. The remaining energy-dependent factors were now defined on single energy plane which does not have any branching points anymore. For these energydependent functions, a system of first-order differential equations is obtained. Then, using the fact that the functions are analytic, they were expanded in the power series to obtain a system of differential equations that determine the expansion coefficients. When the expansion coefficients are obtained for the expansion around the energy E0 = 0,the coefficients are then used to calculate the effective range parameters. For the same hypernuclear systems, the scattering lengths, effective radii, and the other effective-range parameters (up to the order ∼ k8) for the angular momentum ℓ = 0, 1, 2 are calculated. Possible bound and resonant states of the multi-lambda systems ΛΛ(0+), ΛΛΛ(12 −) and ΛΛΛΛ(0+, 1+, 2+) are sought as zeros of the corresponding Jost functions calculated within the framework of the hyperspherical approach with local two-body S-wave potentials describing the ΛΛ interactions. Bound ΛΛ(0+), ΛΛΛ(1 2 −) and states only appears if the two-body potentials are multiplied by a minimum factor of ∼ 1.461 and 3.449. For ΛΛΛΛ(0+, 1+, 2+) systems the bound states appear when the two-body potentials aremultiplied by the factors ∼ 3.018, 4.360 and 3.419. A method for deducing the two-body potential from a given two- or three-body wave function is suggested. This method makes it possible to numerically obtain an unknown potential acting between the particles A and B when we know the potentials of their interaction with a third particle C and know the characteristics of the three-body bound state (ABC). The systems (nnp) and (ΛΛα) were used to show that a three-body wave functions can be constructed using the knowledge of the binding energies and sizes of these systems to deduce reasonable and realistic nn and ΛΛ potentials.