Nuclear three-body problem in the complex energy plane: Complex-scaling Slater method

Abstract

Background:The physics of open quantum systems is an interdisciplinary area of research. The nuclear "openness" manifests itself through the presence of the many-body continuum representing various decay, scattering, and reaction channels. As the radioactive nuclear beam experimentation extends the known nuclear landscape toward the particle drip lines, the coupling to the continuum space becomes exceedingly more important. Of particular interest are weakly bound and unbound nuclear states appearing around particle thresholds. Theories of such nuclei must take into account their open quantum nature. Purpose:To describe open quantum systems, we introduce a complex-scaling (CS) approach in the Slater basis. We benchmark it with the complex-energy Gamow shell model (GSM) by studying energies and wave functions of the bound and unbound states of the two-neutron halo nucleus 6 He viewed as an α + n + n cluster system. Methods: Both CS and GSM approaches are applied to a translationally invariant Hamiltonian with the two-body interaction approximated by the finite-range central Minnesota force. In the CS approach, we use the Slater basis, which exhibits the correct asymptotic behavior at large distances. To extract particle densities from the back-rotated CS solutions, we apply the Tikhonov regularization procedure, which minimizes the ultraviolet numerical noise. Results: We show that the CS-Slater method is both accurate and efficient. Its equivalence to the GSM approach has been demonstrated numerically for both energies and wave functions of 6 He. One important technical aspect of our calculation was to fully retrieve the correct asymptotic behavior of a resonance state from the complex-scaled (square-integrable) wave function. While standard applications of the inverse complex transformation to the complex-rotated solution provide unstable results, the stabilization method fully reproduces the GSM benchmark. We also propose a method to determine the smoothing parameter of the Tikhonov regularization. Conclusions:The combined suite of CS-Slater and GSM techniques has many attractive features when applied to nuclear problems involving weakly bound and unbound states. While both methods can describe energies, total widths, and wave functions of nuclear states, the CS-Slater method-if it can be applied-can provide additional information about partial energy widths associated with individual thresholds.

Structure and decays of nuclear three-body systems: The Gamow coupled-channel method in Jacobi coordinates

“…This is in agreement with the findings of Ref. [34] based on the complex scaling method with COSM coordinates, who obtained the g.s. energy 6 He that was slightly less bound as compared to results of Ref.…”

Calculation of Expectation Values of Operators in the Complex Scaling Method

“…As we see, for a range of Tikhonov parameters the strength function (12) calculated with the regularized Gamow states is identical to the strength function using formula (11) and a similar situation was also encountered in rms radius calculation. As a matter of fact, in [48] a criterion that eliminates too low and too large values of κ was established, confirming that there is a plateau of regularization parameters for which results are converged. This is important to know as we may not have an "exact" reference curve to our disposal.…”

“…As it was shown in [48] and also presented here in Fig. 2, the direct backrotation is very unstable and retrieving the Gamow character of the wavefunction in this way is not possible.…”

Probing resonances in the Dirac equation with quadrupole-deformed potentials with the complex momentum representation method

“…Moreover, the complex scaled equation can be solved by using the boundstate methods, in which the bound states and resonant states are processed equally. These advantages enable the application of CSM to different theoretical frameworks, including the combinations with the few-body models [14], shell models [15,16], and Hartree-Fock theories [17,18]. More applications can be found in Refs.…”