Energy resolution with the Lorentz integral transform

A brief outline of the Lorentz integral transform (LIT) method is given. The method is well established and allows treatment of reactions into the many-body continuum with bound-state-like techniques. The energy resolution that can be achieved is studied by means of a simple two-body reaction. From the discussion it will become clear that the LIT method is an approach with a controlled resolution and that there is no principle problem to even resolve narrow resonances in the many-body continuum. As an example, the isoscalar monopole resonance of ${}^4\mathrm{He}$ is considered. The importance of the choice of a proper basis for the expansion of the LIT states is pointed out. By employing such a basis, a width of 180(70) keV is found for the ${}^4\mathrm{He}$ isoscalar monopole resonance when using a simple central nucleon-nucleon potential model.

Figure 1

Spectrum of the Hamiltonian eigenenergies for the ${}^3{P}_1$ NN channel with the AV18 NN potential for the basis of Eqs. (13) and (14) with various values of $N$ and $b$: $N=10$ with $b=1$, 0.5, and 0.25 fm in panel (a) and $b=0.5$ fm with $N=10$, 30, and 50 in panel (b).

Figure 2

LIT $L\left(\sigma \right)$ of the ${}^3{P}_1$ channel with ${\sigma }_I=10$ MeV and $b=0.5$ fm. (a) $N=10$ and 50; and (b) the ratio $r$ of the LIT with $N=10$, 20, 30, and 40 to the LIT with $N=50$ (the values 0.995 and 1.005 are illustrated by full lines).

Figure 3

LIT $L\left(\sigma \right)$ of the ${}^3{P}_1$ channel with ${\sigma }_I=0.1$, 1, 2.5, and 10 MeV in all four panels ($b=0.5$ fm), but different $N$ values in the various panels: 10 (a), 20 (b), 30 (c), and 50 (d).

Figure 4

LIT $L\left(\sigma \right)$ of $M(q,\omega )$ at $q=300$ MeV/c (TN potential) of ${}^4\mathrm{He}$ with four-body CHH basis from Ref. [8].

Figure 5

Spectrum of the Hamiltonian eigenenergies for the ${}^4\mathrm{He}(J^{\pi }=0^{+})$ states with the TN NN potential for the basis described in the text with $N_4$ basis functions for the radial single-particle basis.

Figure 6

LIT $L\left(\sigma \right)$ of $M(q,\omega )$ at $q=250$ MeV/c (TN potential) of ${}^4\mathrm{He}$ with ${\sigma }_I=0.01$, 0.1, 0.25, 1, and 5 MeV calculated with the new basis states [three-body CHH basis times single-particle basis of Eqs. (17) and (18)]: in panel (a) the single-particle basis with 20 and in panel (b) with 21 states (see text).