Complex-scaled spectrum completeness for pedestrians

Giraud, B. G.; Katō, Kiyoshi

doi:10.1016/s0003-4916(03)00134-9

Cited by 27 publications

(34 citation statements)

References 11 publications

(25 reference statements)

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“…The special cases j = 1 and j = N give the argument η ≡ ArgP as η = θ + π/2 and η = θ, respectively. This was already known from [5]. The function sin 2η + sin(4θ − 2η), see Eq.…”

Section: B Pseudomomentum Planementioning

confidence: 83%

“…1. As in [5], the lowest channel is represented by the heavy, shoulder shaped line, that starts from −∞ on the real P axis, bends up, then backs into the origin P = 0, where it terminates with a slope 2θ. Along the curve, k is real and runs from −∞ to +∞, covering both rims of the initial cut.…”

Section: B Pseudomomentum Planementioning

confidence: 99%

“…For the one channel case, convincing arguments have been advanced a long time ago [4] to prove that this resolution exists. More recently [5], a detailed investigation of the case of two channels, coupled by straightforward potentials, generated a contour integration of the usual Green's function which provided the identity resolution. The task was made reasonably easy by the small complication of the Riemann surface in that case.…”

Section: Introduction Notationsmentioning

confidence: 99%

“…This slight change of representation changes nothing to the physics, obviously. For trivial technical reasons [5], we normalize energy units so that E * N = 4. Also we shall use a short notation, k ≡ k 1 and K ≡ k N .…”

Section: Introduction Notationsmentioning

confidence: 99%

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Complex scaled spectrum completeness for coupled channels

Giraud¹,

Katō²,

Ohnishi³

2004

J. Phys. A: Math. Gen.

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The Complex Scaling Method (CSM) provides scattering wave functions which regularize resonances and suggest a resolution of the identity in terms of such resonances, completed by the bound states and a smoothed continuum. But, in the case of inelastic scattering with many channels, the existence of such a resolution under complex scaling is still debated. Taking advantage of results obtained earlier for the two channel case, this paper proposes a representation in which the convergence of a resolution of the identity can be more easily tested. The representation is valid for any finite number of coupled channels for inelastic scattering without rearrangement.

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Section: B Pseudomomentum Planementioning

confidence: 83%

Section: B Pseudomomentum Planementioning

confidence: 99%

Section: Introduction Notationsmentioning

confidence: 99%

Section: Introduction Notationsmentioning

confidence: 99%

See 2 more Smart Citations

Complex scaled spectrum completeness for coupled channels

Giraud¹,

Katō²,

Ohnishi³

2004

J. Phys. A: Math. Gen.

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“…1). The continuum level density (CLD) ∆(E) is expressed as a balance between the density of states ρ(E) obtained from the Hamiltonian H and the density of continuum states, ρ 0 (E), obtained from the asymptotic Hamiltonian H 0 in the form 11) whereρ(E) is defined through subtraction of the bound state term from ρ(E). Physically, ∆(E) represents the density of unbound levels, which result from the interaction with a finite range.…”

Section: §1 Introductionmentioning

confidence: 99%

Level density in complex scaling method

Suzuki¹,

Myo²,

Katō³

2005

AIP Conference Proceedings

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It is shown that the continuum level density (CLD) at unbound energies can be calculated with the complex scaling method (CSM), in which the energy spectra of bound states, resonances and continuum states are obtained in terms of L 2 basis functions. In this method, the extended completeness relation is applied to the calculation of the Green functions, and the continuum-state part is approximately expressed in terms of discretized complex scaled continuum solutions. The obtained result is compared with the CLD calculated exactly from the scattering phase shift. The discretization in the CSM is shown to give a very good description of continuum states. We discuss how the scattering phase shifts can inversely be calculated from the discretized CLD using a basis function technique in the CSM. §1. IntroductionRecently, there has been much interest in nuclear structures of unstable nuclei, in which exotic nuclear structures have been revealed through the development of radioactive nuclear beam experiments. 1) It has been shown that for such nuclei, for example the so-called neutron halo nuclei, there are extremely weak binding ground states, and most of the excited states are in the continuum energy region. Therefore, to understand the exotic structures and excitations of these nuclei, it is necessary to study continuum and resonant states in unbound energy regions.The continuum level density (CLD) is expected to play an important role in relating experimental data and theoretical models for unbound states. Recently, Kruppa and Arai 2)-4) proposed an interesting method to calculate the CLD and argued that resonance parameters can be determined from the CLD. They start their investigation from the definition of the CLD, 5)where the full and free Green functions are given by G(E) = (E − H) −1 and G 0 (E) = (E − H 0 ) −1 , respectively. Because the Hamiltonian H includes finite range interactions in addition to the asymptotic Hamiltonian H 0 , the CLD expresses the effect from the interactions. When the eigenvalues (ǫ i and ǫ j 0 , respectively) of H and H 0 are obtained approximately within a framework including a finite number (N )

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Techniques to Treat the Continuum Applied to Electromagnetic Transitions in 8Be

2014

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Bremsstrahlung emission in collisions between charged nuclei is equivalent to nuclear gamma decay between continuum states. The way the continuum spectrum can be treated is not unique, and efficiency and accuracy of cross section calculations depend on the chosen method. In this work we describe, relate, and compare three different methods in practical calculations of inelastic cross sections, that is, by (i) treating the initial and final states as pure continuum states on the real energy axis, (ii) discretizing the continuum states on the real energy axis with a box boundary condition, and (iii) complex rotation of the hamiltonian (complex scaling method). The electric quadrupole transitions, 2 + → 0 + and 4 + → 2 + , in α + α scattering are taken as an illustration.Keywords First keyword · Second keyword · More IntroductionThe emission of bremsstrahlung in a collision between two charged particles constitutes an important background effect in Coulomb deexcitation processes. In a classical picture, this phenomenon is understood as the energy radiated due to the deceleration of a charged particle when deflected by another charged particle. The radiated energy is just the kinetic energy lost in the deceleration process. In a quantum mechanical picture, the process can be seen as the γ-emission due to the decay of a two-body system from some two-body continuum state into another continuum state of lower energy. A detailed derivation of the cross section for this kind of processes can be found in [1].The fact that these transitions involve continuum structures, both in the initial and final states, introduce technical as well as conceptual difficulties in the calculations. First, the treatment of the continuum spectrum itself, which can be handled in different ways, but in numerical studies almost necessarily by some kind of discretization. Second, the unavoidable matrix elements between the continuum states involved in the calculation are diverging and thus not well defined without some regularization prescription. These two problems, how to treat the continuum and how to obtain converged results, are possible sources of uncertainty that deserves detailed investigations.Different methods were previously used to compute the bremsstrahlung cross sections for various combinations of nuclei. In [2] α − α collisions were investigated in connection with the resonant electric

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Complex-scaled spectrum completeness for pedestrians

Cited by 27 publications

References 11 publications

Complex scaled spectrum completeness for coupled channels

Complex scaled spectrum completeness for coupled channels

Level density in complex scaling method

Techniques to Treat the Continuum Applied to Electromagnetic Transitions in 8Be

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