Complex Scaling Method in Quantum Systems

• Complex Scaling Method is a transformation that rotates spatial coordinates and momenta into the complex plane, converting resonant states into square-integrable eigenstates for unified treatment of bound and continuum states.
• It employs basis expansions—such as harmonic oscillator and Gaussian bases—to represent the non-Hermitian Hamiltonian and isolate resonance poles effectively.
• Applications span nuclear structure, scattering analyses, and hadron spectroscopy by simplifying the computation of observables and systematically handling resonant and continuum contributions.

Searching arXiv for recent and foundational papers on the Complex Scaling Method to support the article. The complex scaling method (CSM), also called complex rotation, is a non-unitary similarity transformation in which coordinates and momenta are rotated into the complex plane, typically as $r \to r e^{i\theta}$ and $p \to p e^{-i\theta}$. For a Hamiltonian $H=T+V$, the transformed operator $H_\theta = U(\theta)HU(\theta)^{-1}$ converts resonant states into square-integrable eigenstates with complex energies $E = E_r - i\Gamma/2$. In the Aguilar–Balslev–Combes framework, bound-state eigenvalues remain invariant, the non-resonant continuum is rotated into the lower half-plane by $2\theta$, and resonance poles appear as isolated discrete eigenvalues. CSM therefore places bound, resonant, and discretized scattering states in a common $L^2$ setting and replaces explicit outgoing-wave asymptotics by decaying asymptotics suitable for basis expansion and matrix diagonalization (Papadimitriou et al., 2014).

1. Spectral foundation

At the operator level, CSM is defined by the transformation

$U(\theta): r \rightarrow r e^{i\theta},$

with the induced momentum rotation $p \to p e^{-i\theta}$. For local Hamiltonians, the kinetic term scales as $T \to e^{-2 i\theta}T$, while the potential becomes $p \to p e^{-i\theta}$0. The complex-scaled Schrödinger equation,

$p \to p e^{-i\theta}$1

has a non-Hermitian spectral structure that is central to the method (Liu et al., 2012).

The standard spectral consequences are threefold. First, bound states remain on the negative real axis and are independent of $p \to p e^{-i\theta}$2. Second, resonance poles of the Green’s function or $p \to p e^{-i\theta}$3-matrix appear as discrete eigenvalues $p \to p e^{-i\theta}$4, ideally stationary with respect to $p \to p e^{-i\theta}$5. Third, the continuum branch cut is rotated downward by $p \to p e^{-i\theta}$6, so continuum eigenvalues align along a ray in the complex energy plane. In practical calculations, this produces a characteristic separation into bound states, isolated resonance points, and strings of discretized continuum points (Liu et al., 2012).

This reformulation is especially consequential because outgoing Gamow asymptotics are transformed into decaying asymptotics. A resonant wave function that is not square-integrable before scaling becomes representable in a finite $p \to p e^{-i\theta}$7 basis after scaling. In that sense, CSM is not merely a pole-finding device; it is a spectral reorganization that turns resonance physics into a generalized eigenvalue problem.

2. Operator formulations and basis machinery

The most direct implementations use basis expansions. In axially deformed nuclear mean fields, one expands the complex-scaled wave function in a spherical harmonic-oscillator basis,

$p \to p e^{-i\theta}$8

and obtains a complex symmetric matrix representation of $p \to p e^{-i\theta}$9. For deformed Woods–Saxon potentials, the central term remains diagonal in angular momentum, the quadrupole term mixes partial waves through $H=T+V$0, and the spin–orbit term couples orbital and spin degrees of freedom while conserving $H=T+V$1 and parity. In the $H=T+V$2Ne application, $H=T+V$3 major harmonic-oscillator shells were used, with stability reached for $H=T+V$4 and optimal rotation angles around $H=T+V$5–$H=T+V$6 (Liu et al., 2012).

For realistic non-local nucleon–nucleon interactions, direct substitution $H=T+V$7 is cumbersome. A technically important alternative is to leave the interaction unchanged and instead complex-scale the basis functions, equivalently promoting the harmonic-oscillator length $H=T+V$8 to $H=T+V$9. In this representation the Hamiltonian remains non-Hermitian, but the matrix elements of non-local interactions such as JISP16, chiral N$H_\theta = U(\theta)HU(\theta)^{-1}$0LO$H_\theta = U(\theta)HU(\theta)^{-1}$1, and SRG-evolved N$H_\theta = U(\theta)HU(\theta)^{-1}$2LO can be computed in a uniform way, and the resulting complex symmetric matrix can be diagonalized directly (Papadimitriou et al., 2014).

Other realizations use Gaussian bases, cluster orbital shell-model bases, polynomial–Gaussian bases, or finite elements. In five-body and four-body cluster calculations, Gaussian expansions accommodate both compact core–valence structure and extended resonance tails. In black-hole perturbation theory, polynomial $H_\theta = U(\theta)HU(\theta)^{-1}$3 Gaussian bases are used after complex scaling of the tortoise coordinate. In anisotropic resonance problems, radial rather than Cartesian complex scaling leads to a Fredholm operator and convergent finite-element approximations under simultaneous domain truncation and discretization [(Myo et al., 2012); (Ogawa et al., 5 May 2026); (Halla, 2021)].

Across these realizations, the common computational pattern is the same: CSM trades asymptotic matching for analytic continuation, and then solves a complex, non-Hermitian, but finite-dimensional spectral problem.

3. Observables, phase shifts, and scattering formulations

Beyond locating poles, CSM supports direct construction of scattering observables. A standard object is the complex-scaled continuum level density,

$H_\theta = U(\theta)HU(\theta)^{-1}$4

which is related to the derivative of the phase shift with respect to energy. In two-body applications this enables a decomposition of scattering phase shifts and cross sections into resonant and non-resonant continuum contributions. Narrow resonances generate step-like contributions of approximately $H_\theta = U(\theta)HU(\theta)^{-1}$5 to the phase shift, while the continuum component provides a smooth background; their interference produces Fano-like line shapes in partial cross sections (Odsuren et al., 2014).

A complementary development treats scattering directly through driven equations. For few-body scattering, one writes the total wave function as incoming plus scattered parts and complex-scales the inhomogeneous equations. Because outgoing waves become exponentially damped after scaling, the unknown scattered components satisfy trivial boundary conditions and can be solved with bound-state techniques. This strategy has been implemented for two-body, three-body, and four-body scattering, including elastic, rearrangement, and breakup reactions, without explicit construction of asymptotic scattering boundary conditions [(Lazauskas et al., 2011); (Lazauskas et al., 2012)].

Expectation values and transition strengths raise an additional subtlety. In the standard formulation one evaluates

$H_\theta = U(\theta)HU(\theta)^{-1}$6

with the operator rotated as $H_\theta = U(\theta)HU(\theta)^{-1}$7. A distinct line of work instead reconstructs the Gamow asymptotic character of the state by backrotation of the CSM wave function. Direct backrotation is numerically ill-posed, but Tikhonov regularization stabilizes the inverse transformation and permits expectation values to be computed with the unrotated operator on a regularized Gamow-like state. For radii and dipole strengths, the results agree with standard CSM expectation values over a plateau of regularization parameters, while restoring a more direct connection to the original resonance wave function (Papadimitriou, 2015).

4. Nuclear structure and reaction applications

In nuclear structure, CSM has been extended from spherical to axially deformed mean fields. For a single neutron in an axially symmetric quadrupole-deformed Woods–Saxon potential, the good quantum numbers are parity and the projection $H_\theta = U(\theta)HU(\theta)^{-1}$8 of total angular momentum on the symmetry axis. In $H_\theta = U(\theta)HU(\theta)^{-1}$9Ne, the method reproduces bound and resonant energies from the multichannel scattering approach and reveals a deformation-dependent evolution of widths: the $E = E_r - i\Gamma/2$0 and $E = E_r - i\Gamma/2$1 resonances decrease in energy with increasing $E = E_r - i\Gamma/2$2, but their widths exhibit a non-monotonic behavior, increasing away from spherical shape, reaching a maximum at intermediate deformation, and then decreasing again. The $E = E_r - i\Gamma/2$3 level becomes weakly bound around $E = E_r - i\Gamma/2$4, and the interval $E = E_r - i\Gamma/2$5, with an optimal deformation around $E = E_r - i\Gamma/2$6, was identified as favorable for a deformed halo in $E = E_r - i\Gamma/2$7Ne (Liu et al., 2012).

Cluster models provide a distinct many-body realization. In $E = E_r - i\Gamma/2$8C, treated as $E = E_r - i\Gamma/2$9, CSM yields the ground state as a five-body resonance with $2\theta$0 MeV and $2\theta$1 MeV relative to the $2\theta$2He$2\theta$3 threshold, and predicts a broad excited $2\theta$4 resonance at $2\theta$5 MeV with $2\theta$6 MeV. The ground state is dominated by the $2\theta$7 configuration, while the $2\theta$8 state is dominated by $2\theta$9, and CSM makes it possible to compute occupation numbers, pair numbers, and radii for these Gamow states within a single diagonalization (Myo et al., 2012).

In $L^2$0B, described as $L^2$1He$L^2$2, CSM predicts five resonances and permits spectroscopic factors to be defined for $L^2$3Be$L^2$4 components using bi-orthogonal matrix elements. The ground state shows a particularly strong $L^2$5Be$L^2$6 component, larger than the mirror $L^2$7He$L^2$8 component in $L^2$9He, indicating mirror-symmetry breaking driven by the Coulomb-shifted proximity of the $U(\theta): r \rightarrow r e^{i\theta},$0Be$U(\theta): r \rightarrow r e^{i\theta},$1 threshold (Myo et al., 2011).

The same framework has been carried into reaction calculations above breakup thresholds. In three- and four-body configuration-space scattering, complex scaling of Faddeev and Faddeev–Yakubovsky equations converts outgoing asymptotics into exponentially damped functions and allows elastic and breakup observables to be computed with trivial boundary conditions. Applications were reported for $U(\theta): r \rightarrow r e^{i\theta},$2–$U(\theta): r \rightarrow r e^{i\theta},$3, $U(\theta): r \rightarrow r e^{i\theta},$4–$U(\theta): r \rightarrow r e^{i\theta},$5, and $U(\theta): r \rightarrow r e^{i\theta},$6H scattering, including Hamiltonians with short-range forces and repulsive Coulomb interaction (Lazauskas et al., 2012).

5. Extensions across physics

A major extension beyond traditional resonance spectroscopy is the use of CSM with realistic, strong, non-local nucleon–nucleon interactions. In the proton–neutron system, the deuteron bound state remains invariant under complex rotation, while NN scattering phase shifts in uncoupled $U(\theta): r \rightarrow r e^{i\theta},$7 channels can be extracted from the same complex-scaled spectrum through the continuum level density. Calculations with JISP16, chiral N$U(\theta): r \rightarrow r e^{i\theta},$8LO$U(\theta): r \rightarrow r e^{i\theta},$9, and SRG-evolved N$p \to p e^{-i\theta}$0LO show that phase shifts become practically indistinguishable from exact results արդեն at $p \to p e^{-i\theta}$1 rad, demonstrating that the ABC-type spectral picture persists for realistic non-local interactions (Papadimitriou et al., 2014).

In hadron spectroscopy, CSM has been applied to meson–baryon and meson–meson systems with one-boson-exchange interactions. For the double-charm system $p \to p e^{-i\theta}$2, treated as a $p \to p e^{-i\theta}$3 molecule with $p \to p e^{-i\theta}$4, the method yields a quasibound state with binding energy $p \to p e^{-i\theta}$5 keV relative to the $p \to p e^{-i\theta}$6 threshold and width $p \to p e^{-i\theta}$7 keV, with dominant $p \to p e^{-i\theta}$8-wave $p \to p e^{-i\theta}$9 and $T \to e^{-2 i\theta}T$0 components of $T \to e^{-2 i\theta}T$1 and $T \to e^{-2 i\theta}T$2, respectively. For $T \to e^{-2 i\theta}T$3, treated as a $T \to e^{-2 i\theta}T$4 molecule with $T \to e^{-2 i\theta}T$5, the method gives a quasibound state at $T \to e^{-2 i\theta}T$6 keV relative to the $T \to e^{-2 i\theta}T$7 threshold with width $T \to e^{-2 i\theta}T$8 keV, dominated by the neutral $T \to e^{-2 i\theta}T$9-wave component at $p \to p e^{-i\theta}$00 (Cheng et al., 2022).

CSM has also migrated into black-hole perturbation theory. In four-dimensional Schwarzschild–de Sitter spacetimes, complex scaling of the tortoise coordinate converts the outgoing-wave quasinormal-mode boundary-value problem into a non-Hermitian spectral problem. Quasinormal-mode poles then appear as discrete eigenvalues, while the rotated continuum can be analyzed through a continuum level density. This unifies pole and continuum sectors for scalar, electromagnetic, and gravitational perturbations and extends, at least formally, to higher-dimensional tensor- and vector-type sectors (Ogawa et al., 5 May 2026).

In non-Hermitian quantum mechanics, CSM has been used to study the broken PT-symmetry phase of the Swanson Hamiltonian, where discrete complex energies, bi-orthogonality, response functions, Wigner functions, and a continuity equation can be constructed within the complex-scaled representation. A further development concerns exceptional points in genuine scattering systems: in a one-dimensional model, resonance poles of the $p \to p e^{-i\theta}$01-matrix are realized as eigenvalues of a dilated Hamiltonian, enabling analysis of self-orthogonality and Berry phase when a resonance becomes embedded in the continuum spectrum (Fernández et al., 2024, Morikawa et al., 31 Dec 2025).

6. Numerical subtleties, misconceptions, and current directions

CSM’s practical effectiveness depends on several nontrivial choices. Finite $p \to p e^{-i\theta}$02 bases discretize the continuum, so the ideal $p \to p e^{-i\theta}$03-independence of resonance eigenvalues is only approximate. In $p \to p e^{-i\theta}$04Ne, resonance energies trace short trajectories in the complex plane as $p \to p e^{-i\theta}$05 varies, and the optimal estimate is taken at the point where $p \to p e^{-i\theta}$06 is minimal. Similar basis-size and angle studies are required in other implementations, especially for broad or near-threshold states (Liu et al., 2012).

A common misconception is that CSM is restricted to local short-range potentials. The method has been implemented for realistic non-local NN interactions by shifting the complex scaling from the potential kernel to the basis functions, and for Coulomb-bearing scattering problems through Coulomb-adapted driven equations and continuum-level-density constructions. At the same time, some sectors remain numerically delicate: the $p \to p e^{-i\theta}$07 NN channel, with its nearby virtual state, was found to require much larger harmonic-oscillator spaces for accurate CLD-based phase shifts (Papadimitriou et al., 2014).

Another misconception is that any coordinate scaling is physically harmless. For anisotropic media this is not the case: Cartesian complex scaling can impose an unphysical radiation condition and lead to erroneous and unstable results. Radial complex scaling, by contrast, was shown to avoid this drawback in scalar anisotropic resonance problems, yielding a Fredholm operator and convergent finite-element approximations under simultaneous truncation and discretization (Halla, 2021).

Current developments are increasingly algorithmic. A recent scattering emulator uses CSM to construct a single reduced basis across multiple partial waves and optical-potential parameters, with stable performance and no anomalies of the kind encountered in some alternative emulation techniques. Demonstrations for $p \to p e^{-i\theta}$08Ca and $p \to p e^{-i\theta}$09Be$p \to p e^{-i\theta}$10Zn show that the square-integrable, channel-uniform structure produced by complex scaling is compatible with reduced-order modeling and high-throughput reaction analysis (Liu et al., 2024).

Taken together, these results support a broad characterization of CSM: it is a spectral method for analytic continuation that replaces explicit outgoing-wave boundary conditions by decaying $p \to p e^{-i\theta}$11 asymptotics, thereby linking resonance theory, continuum response, and non-Hermitian operator analysis within a single computational framework.