Improving the accuracy of the Lorentz Integral Transform method using complex kernels

Technical Summary: Improving the Accuracy of the Lorentz Integral Transform Method Using Complex Kernels

Problem Statement
The Lorentz Integral Transform (LIT) method is a established technique for calculating quantum response functions $R(\omega)$ in nuclear, atomic, and condensed-matter physics. It circumvents the difficulty of explicitly constructing continuum wavefunctions by computing a convolution of the response with a Lorentzian kernel, $L(\sigma, \Gamma)$. However, recovering the physical response $R(\omega)$ from the transform $L(\sigma, \Gamma)$ requires inverting an ill-posed problem. Features in $R(\omega)$ narrower than the kernel width $\Gamma$ are exponentially suppressed, and their reconstruction amplifies errors in $L$. While the inversion itself is manageable, the trade-off between resolution (requiring small $\Gamma$) and the numerical difficulty of solving the LIT equation (which becomes harder as $\Gamma \to 0$) limits the method's applicability to heavier systems. Previous efforts have focused on improving the inversion algorithms (e.g., basis expansions, regularization, maximum entropy), but the question of whether the kernel itself could be chosen to improve the conditioning of the inversion has remained largely unexplored.

Methodology
The authors propose that the wavefunctions $|\tilde{\Psi}(\sigma)\rangle$ generated in any standard LIT calculation contain sufficient information to evaluate two additional integral transforms with superior conditioning properties, at negligible additional computational cost.

1. Derivation of Alternative Transforms:
   Starting from the standard LIT wavefunction defined by $(\hat{H} - E_0 - \sigma + i\Gamma)|\tilde{\Psi}(\sigma)\rangle = |\Phi\rangle$, where $|\Phi\rangle = \hat{O}|\Psi_0\rangle$, the authors derive:

   • Complex Stieltjes Transform ($L_{CS}$): Evaluated as the overlap $-\frac{1}{\Gamma}\langle\Phi|\tilde{\Psi}(\sigma)\rangle$. Its kernel, $K_{CS}$, has a pole at $\sigma - i\Gamma$. Its imaginary part reproduces the standard Lorentzian, while its real part provides antisymmetric, sign-sensitive information.
   • Double-Pole Transform ($L_2$): Evaluated as $-\langle\tilde{\Psi}(\sigma, -\Gamma)|\tilde{\Psi}(\sigma, \Gamma)\rangle$. This requires solving the LIT equation with the opposite sign of $i\Gamma$ (or using time-reversal symmetry to relate the solutions). Its kernel, $K_2$, features a double pole at $\sigma - i\Gamma$.

2. Fourier Analysis of Stability:
   The authors analyze the deconvolution problem in Fourier space. Since the integral transform is a convolution, the inversion involves dividing the Fourier transform of the data by the Fourier transform of the kernel, $\tilde{K}(k)$. Stability is determined by how slowly $|\tilde{K}(k)|$ decays at large $k$.

   • All kernels share the same exponential decay rate $e^{-\Gamma k}$ determined by the pole distance $\Gamma$.
   • However, the Complex Stieltjes kernel has one-sided Fourier support (vanishing for $k<0$), allowing more information to pass through compared to the symmetric Lorentzian.
   • The Double-Pole kernel's Fourier transform includes an additional factor of $k$, causing it to decay more slowly than the Stieltjes kernel at high frequencies.
   • This analysis predicts a stability ordering: $K_2 \succ K_{CS} \succ K_{LIT}$.

3. Numerical Benchmark:
   The method is tested on the deuteron photodisintegration problem using a pionless effective-field-theory potential. Exact LIT wavefunctions are computed, and controlled multiplicative complex Gaussian noise is added to simulate stochastic or truncation errors. The response functions are inverted using a regularized basis-expansion procedure for all four cases: Standard LIT, LIT-GS (an overlap-based estimator of the standard LIT), Complex Stieltjes, and Double-Pole.

Key Contributions and Results

• Novel Kernels: The paper demonstrates that the standard LIT wavefunctions can be repurposed to compute Complex Stieltjes and Double-Pole transforms without solving new many-body problems, requiring at most one additional LIT equation solve (often avoided via symmetry).
• Quantitative Improvement: In the presence of noise, the alternative kernels yield response functions with significantly reduced scatter compared to the standard LIT:
  • The Complex Stieltjes kernel reduces noise-induced scatter by a factor of 2–3.
  • The Double-Pole kernel reduces noise-induced scatter by a factor of 3–4.
• Mechanism of Improvement: The improvement arises from two mechanisms:
  1. Linear Error Propagation: Using the overlap with the source state (linear in $|\tilde{\Psi}\rangle$) rather than the norm (quadratic in $|\tilde{\Psi}\rangle$) eliminates positive bias and reduces error amplification.
  2. Better-Conditioned Kernels: The specific spectral properties of the Stieltjes and Double-Pole kernels make the deconvolution problem better conditioned, particularly at high frequencies where noise amplification is most severe.
• Robustness: The stability ordering ($K_2 \succ K_{CS} \succ K_{LIT}$) holds consistently across noise levels spanning two orders of magnitude.

Significance and Claims
The authors claim that this approach offers a substantial improvement in the accuracy and stability of LIT calculations with minimal computational overhead. Because the improvement is achieved through post-processing of existing wavefunctions, it can be immediately adopted in existing LIT frameworks (e.g., hyperspherical harmonics, coupled-cluster, neural-network quantum states).

The primary practical significance is that the enhanced inversion stability allows researchers to work with larger values of $\Gamma$ for a fixed accuracy target. Since the difficulty of solving the LIT equation increases as $\Gamma$ decreases (due to the growing spatial extent of the solution), using a larger $\Gamma$ eases the numerical burden. This could potentially extend the reach of LIT methods to heavier nuclear systems that are currently limited by the trade-off between resolution and computational cost. The paper does not propose new experimental applications but frames the method as a robust enhancement to the theoretical toolkit for calculating response functions.