Complex Scaling Method (CSM)

• Complex Scaling Method is a quantum mechanics technique that transforms divergent resonance (Gamow) states into square-integrable functions via complex coordinate rotation.
• It rotates the continuum spectrum to isolate resonant poles as discrete eigenvalues in the complex energy plane, enabling precise computation of resonance parameters.
• The method supports various L²-basis expansion strategies and numerical techniques for accurately extracting observables in scattering and nuclear resonance studies.

The complex scaling method (CSM) is a similarity transformation technique in quantum mechanics that enables rigorous treatment of resonant and continuum states, particularly within non-Hermitian formulations. By analytically continuing spatial coordinates into the complex plane, CSM transforms exponentially divergent resonance (Gamow) wave functions into square-integrable (L²) functions and rotates the continuum spectrum, distinguishing resonant poles as isolated discrete eigenvalues in the complex energy plane. CSM is essential for the computation of resonance parameters, spectroscopic factors, and strength functions in unbound nuclear systems, as well as for practical computations in few-body scattering and quantum resonance spectroscopy.

1. Mathematical Foundations and Theoretical Framework

The core of the complex scaling method is the transformation of all or a subset of coordinates by a complex phase: $r_i \rightarrow r_i\, e^{i\theta}$ where $\theta$ is the complex scaling angle ($0 < \theta < \pi/2$). The procedure systematically transforms the Hamiltonian,

$H_\theta = U(\theta) H U(\theta)^{-1}$

and the corresponding eigenvalue problem,

$H_\theta \Psi_\theta = E_\theta \Psi_\theta.$

The ABC theorem (Aguilar–Balslev–Combes) guarantees that under this transformation:

• Bound-state eigenvalues remain on the negative real axis.
• Scattering and non-resonant continuum eigenvalues rotate into the lower-half complex energy plane by angle $2\theta$.
• Resonant states (with diverging Gamow asymptotics) are isolated as poles at $E_r - i\Gamma/2$ (resonance position and width).

This translation turns resonance calculations into a standard L²-type eigenvalue problem, supporting the use of bound-state computational technology in open quantum systems (Myo et al., 2010, Myo et al., 2014).

2. Many-Body Applications: COSM+CSM and Resonance Spectroscopy

In the paper of weakly bound and unbound nuclei, the CSM is frequently employed in concert with the cluster orbital shell model (COSM). Here, the nucleus is represented as an inert core with several valence nucleons, and the total wave function is a superposition of configuration states

$\Psi^J(A) = \sum_c C_c^J \Phi_c^J(A)$

built by acting creation operators on the core.

CSM enables imposition of correct asymptotic (scattering) boundary conditions for resonances. Diagonalization of the complex-scaled Hamiltonian yields both resonance poles and continuum eigenvalues in a single computational step, and the resulting L² wave functions can be directly employed to compute observables such as:

• Spectroscopic factors: Evaluated for unbound nuclei as biorthogonal overlaps between Gamow states and cluster channel wave functions. For instance, in $^7$He, S-factors quantify the decomposition of its resonance into $^6$He-n components, revealing dominant contributions from excited $^6$He($2^+$) (Myo et al., 2010).
• Pairing structure analysis: In five-body systems like $^8$He, the decomposition of resonance wave functions elucidates the contributions of specific neutron pairings, e.g., dominance of $2^+$ pairs in the ground state and significant $0^+$, $1^+$, $2^+$ mixing in excited states.

The resonance energies and widths are extracted as discrete points in the complex plane, permitting rigorous comparison with experiment. The CSM-based COSM framework is essential to quantifying decay channel contributions and to revealing mirror symmetry breaking (e.g., in $^7$B vs. $^7$He) driven by Coulomb effects (Myo et al., 2011).

3. Continuum Structure, Strength Functions, and Monopole Transitions

CSM not only isolates resonance poles but also delivers a complete resolution of the continuum for evaluating strength distributions. Using the extended completeness relation, the Green's function is formulated as

$$\mathcal{G}_\theta(E) = \frac{1}{E - H_\theta} = \sum_\nu \frac{|\Psi_\theta^\nu\rangle \langle\tilde\Psi_\theta^\nu|}{E - E_\theta^\nu}$$

which is key for computing strength functions such as monopole response: $$\mathcal{S}(E) = -\frac{1}{\pi} \sum_\nu \operatorname{Im} \left( \frac{\langle \tilde\Psi_\theta^0|O_\theta^\dagger|\Psi_\theta^\nu\rangle \langle\tilde\Psi_\theta^\nu|O_\theta|\Psi_\theta^0\rangle}{E - E_\theta^\nu} \right)$$ This formalism clarifies the dominance of sequential breakup processes (e.g., $^8$He → $^7$He + n → $^6$He + n + n) in the monopole excitation strengths and provides separation between resonance and non-resonance continuum contributions (Myo et al., 2010).

4. Computational Implementation and Basis Expansion Strategies

Taking advantage of the square integrability of all complex-scaled states, CSM supports several efficient discretization schemes:

• L²-basis expansion (e.g., Gaussian or harmonic oscillator): The complex-scaled Hamiltonian is represented in a finite basis, and resonance/continuum states are resolved by matrix diagonalization.
• Complex-range Gaussian or Slater-type basis: For accurate description of oscillatory resonance and continuum states at large scaling angle, complex-range Gaussians $(r^\ell e^{-(1 \pm i\omega)(r/r_n)^2})$ (Ohtsubo et al., 2013) or Slater-type functions $r^n e^{-\alpha r}$ (Kruppa et al., 2013) are employed.
• Tikhonov regularization for back-rotation: Restoration of the physical Gamow asymptotics for expectation value calculations relies on stable numerical inversion, typically accomplished by regularized Fourier inversion schemes (Papadimitriou, 2015, Kruppa et al., 2013).

These implementation choices are dictated by the needs of the system: complex-range bases ameliorate ill-conditioning and permit larger scaling angles for broader or highly oscillatory resonances.

5. Applications in Scattering: Extracting Observables and Benchmarking

CSM provides a unified approach to scattering problems, enabling:

• Direct extraction of phase shifts and amplitudes from discretized spectra (via continuum level densities and Green's function formalism).
• Treatment of both elastic and breakup channels in few-body scattering, with simple boundary conditions replacing intricate asymptotic forms.
• Inclusion of long-range interactions (e.g., Coulomb) by analytic continuation or Faddeev–Merkuriev extensions, allowing accurate phase shifts and cross section calculations for charged-particle collisions (Volkov et al., 2011, Lazauskas et al., 2011, Lazauskas, 2016).

Benchmark studies hold that phase shifts and other observables computed in CSM show excellent agreement with established results, provided the scaling angle and basis are chosen to balance continuum resolution and numerical stability.

6. Generalizations and Limitations

CSM is extensible to:

• Deformed nuclei: The method generalizes to axially symmetric deformed potentials by complex rotation of radial variables and expansion in a harmonic oscillator basis, enabling computation of resonance energies as functions of deformation and the analysis of halo phenomena (Liu et al., 2012).
• Multichannel and coupled-channel systems: With improved contour choices in momentum space, CSM is adapted for extraction of virtual and resonance states, even on anti-physical Riemann sheets (Chen et al., 2023).

Limitations include:

• Analyticity requirements: Potentials must be analytic in the complex rotated domain, restricting scaling angle selection.
• Divergent inhomogeneous terms: Multi-channel and breakup problems require careful handling of the inhomogeneous (incoming) wave, often imposing further limits on the scaling parameter.

Empirically, the scaling angle must be small enough to avoid non-physical oscillations, yet large enough to separate resonance poles from rotated continuum branches.

7. Impact and Future Directions

The CSM has transformed the computational treatment of unbound states in nuclear, hadronic, atomic, and quantum scattering systems. Applications include:

• Extraction of resonance parameters and decay widths in exotic nuclei, hadronic molecules, and few-body systems.
• Computation of observables (e.g., spectroscopic factors, transition strengths, phase shifts) traditionally inaccessible in open quantum systems.
• Extension to problems involving strangeness, open heavy flavor, and reactions with complex spatial asymptotics.

Future research includes expanding multi-channel CSM approaches, enhancing basis set strategies for very broad or near-threshold resonances, and further unification of scattering and bound-state methodology for many-body quantum systems.

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CSM thus provides a mathematically rigorous, computationally accessible, and physically interpretable framework for the paper of resonances and non-resonant continua in quantum systems ranging from light nuclei to complex atomic and molecular problems.