Partial Spectral Functions: Theory & Applications

• Partial Spectral Functions (PSFs) are real-energy functions derived from Lehmann decomposition that isolate particle and hole contributions in quantum many-body systems.
• PSFs serve as foundational components in numerical schemes like the overcomplete intermediate representation, enabling efficient analytic continuation and data compression.
• They play a critical role in solving inverse problems and unmixing in signal processing, despite challenges from overcompleteness and ill-conditioning.

Partial Spectral Functions (PSFs) are fundamental constructs in the spectral analysis of quantum many-body systems and in signal mixture decomposition, serving as building blocks for multipoint Green’s functions and as parametric objects modeling system-dependent spectral responses. PSFs isolate system-specific contributions into real-energy functions, enabling efficient representation, analytic continuation, and unmixing in both theoretical and practical settings. Their structure naturally encodes particle–hole asymmetry and underpins state-of-the-art compression and reconstruction techniques. PSFs are rigorously defined both in the context of Green’s function theory and as parametrized kernels in mixture modeling, with recent advances placing them at the center of numerical schemes for tackling high-dimensional response functions and for addressing inverse problems with non-convex structure (Dirnböck et al., 2024, Michelena et al., 24 Feb 2025).

1. Definition and Mathematical Construction

In quantum many-body theory, Partial Spectral Functions $\rho_{\bar{1}\bar{2}}(\epsilon)$ arise as components of the Lehmann decomposition for multipoint Green’s functions. For two-point (one-particle) Green’s functions, PSFs separately encode the "particle" ($\rho_{12}$) and "hole" ($\rho_{21}$) contributions. For two fermionic operators $A_1$, $A_2$, the time-ordered imaginary-time Green’s function is decomposed as

$$\tilde G(\tau_1,\tau_2) = -\langle T\,A_1(\tau_1)A_2(\tau_2)\rangle = \sum_{\bar{1}\bar{2}}\tilde G_{\bar{1}\bar{2}}(\tau_1,\tau_2),$$

where $\bar{1}\bar{2} \in \{12,21\}$ denotes time-orderings.

The Fourier transform yields

$$G(i\nu_1,i\nu_2) = \sum_{\bar{1}\bar{2}}G_{\bar{1}\bar{2}}(i\nu_1,i\nu_2),$$

with each partial Green’s function (PGF) admitting a spectral representation via its associated PSF: $$G_{\bar{1}\bar{2}}(i\nu_1,i\nu_2) = \beta\;\delta_{i\nu_1+i\nu_2,0}\int_{-W}^{W}\!d\epsilon\,\frac{\rho_{\bar{1}\bar{2}}(\epsilon)}{i\nu_{\bar{1}}-\epsilon}.$$ Explicitly, the PSF is given by

$$\rho_{\bar{1}\bar{2}}(\epsilon) = \mathrm{sgn}(\bar{1}\bar{2}) \sum_{\psi} e^{-\beta E_\psi} \langle\psi|A_{\bar{1}}\,\delta(\epsilon+E_\psi-H)\,A_{\bar{2}}|\psi\rangle,$$

which isolates the system-dependent matrix elements into a single-variable function of real energy $\rho_{12}$0 (Dirnböck et al., 2024).

In mixture model contexts, a parametric family $\rho_{12}$1 is considered, with $\rho_{12}$2 a compact interval of admissible "shape" parameters and $\rho_{12}$3 satisfying smoothness and integrability conditions. The collection $\rho_{12}$4 forms a smooth one-dimensional manifold in $\rho_{12}$5. This formulation enables the use of PSFs as building blocks in the modeling and unmixing of convolved spike signals, with their geometry controlled by derivatives $\rho_{12}$6 for $\rho_{12}$7 (Michelena et al., 24 Feb 2025).

2. Role in Spectral Representation and Analytic Continuation

PSFs are integral to expressing Green’s functions in terms of spectral densities, facilitating analytic continuation and real-frequency evaluation. Starting from the Lehmann representation, the Green’s function is obtained by

$\rho_{12}$8

where each PSF reflects a specific operator ordering. The full spectral function entering the complete Green’s function arises from

$\rho_{12}$9

with

$\rho_{21}$0

Analytic continuation to real frequencies is accomplished by $\rho_{21}$1, yielding retarded Green’s functions evaluated via integrals over PSFs (Dirnböck et al., 2024). This structure provides direct access to dynamic response properties in both computational and theoretical studies.

3. Compression and Numerical Evaluation via Overcomplete Intermediate Representation

The Overcomplete Intermediate Representation (OIR) framework leverages singular-value decompositions of integral kernels to compactly encode PGFs in Matsubara space. The kernel is expanded as

$\rho_{21}$2

This yields a basis expansion

$\rho_{21}$3

with coefficients

$\rho_{21}$4

The two-to-one correspondence arises as only the combinations $\rho_{21}$5 (linked to $\rho_{21}$6) are directly accessible in the full Green’s function. As a result, the OIR basis is formally overcomplete and solutions for the underlying PSFs are non-unique without further regularization (Dirnböck et al., 2024).

Numerical protocols based on OIR/PSF proceed as follows:

1. Collection of dense Matsubara data (up to $\rho_{21}$7; millions of points).
2. Sparse sampling and least-squares fitting (design matrix $\rho_{21}$8 over $\rho_{21}$9 points, coefficient vector of length $A_1$0).
3. Reconstruction in real frequency by term-wise analytic continuation.

Typical compression ratios reach $A_1$1 with in-sample errors $A_1$2. In concrete NRG applications, such as the single-impurity Anderson model, compression and qualitative reconstruction of semi-partial Green’s functions are achieved for complex multi-channel data (Dirnböck et al., 2024).

4. Inverse Problems and Unmixing via Non-Convex Optimization

In nonparametric mixture models with known spike locations, recovery of PSF parameters and amplitudes amounts to nonlinear least-squares estimation over a PSF-manifold: $A_1$3 with $A_1$4 the block-dictionary matrix built from the manifold family $A_1$5 and $A_1$6 the amplitudes. The conditioning of the unmixing problem is characterized by coherence functions $A_1$7 and interference functions $A_1$8, which depend on the minimum spike separation $A_1$9 and measure correlations of shifted (possibly differentiated) PSFs.

A key theoretical result establishes that, under assumptions of Lipschitz continuity for these functions and for $A_2$0, and for sufficiently large sample sizes, the loss landscape exhibits a strong basin of attraction near the ground truth. Explicit bounds on the basin radius and the Hessian’s eigenvalue spectrum are derived from coherence/interference properties and amplitude scales, ensuring local convergence for line-search algorithms such as gradient descent or Gauss–Newton when initialized within the basin (Michelena et al., 24 Feb 2025).

Numerical experiments with Lorentz PSF kernels affirm that the basin shrinks with decreasing spike separation (increased non-convexity), and that theoretical bounds accurately predict practical convergence regimes.

5. Practical Applications and Numerical Case Studies

PSFs enable significant data compression, efficient analytic continuation, and direct physical interpretation in computational quantum many-body physics, as well as robust unmixing in signal processing applications. Key reported metrics include compression ratios exceeding 400–500 at controlled accuracy ($A_2$1), in-sample errors $A_2$2, and out-of-sample errors $A_2$3.

In quantum impurity problems such as the SIAM, OIR/PSF compression has reduced multi-million point data sets to $A_2$4 coefficients with minimal residual error. However, reconstructions of partial Green’s functions provide only qualitative agreement in part of the data channels, illustrating limitations due to overcompleteness and ill-conditioning.

Experimental studies in laser-induced breakdown spectroscopy (LIBS) demonstrate that nonlinear least-squares unmixing of PSFs enables accurate recovery of concentrations and PSF shapes from spectral mixtures. For aluminum-silicon LIBS samples, a fit error (residual vs. preprocessed observation) of approximately 15.33% and relative aluminum concentration error of 0.023% have been reported, validating the ability of the PSF-based framework to perform robust multicomponent quantification (Michelena et al., 24 Feb 2025).

6. Structural Challenges and Open Directions

While the PSF formalism delivers substantial gains in compression and computational tractability, several intrinsic challenges remain:

• Formal overcompleteness of the OIR basis in quantum applications leads to ill-conditioned least-squares problems, with ambiguity in reconstructing the physically correct PSFs. Only semi-partial combinations are determined directly; isolating individual PSFs requires principled regularization.
• Analytic continuation of high-order (e.g., two-particle) quantities remains delicate, as fitted coefficients may deviate from the true spectral structure due to overcompleteness.
• In mixture modeling, conditioning deteriorates as spike separation decreases, reducing convergence basin size and exacerbating the non-convexity of the loss landscape.

Open research questions include:

• Development of regularization schemes to select physically meaningful PSFs from overcomplete or ill-conditioned representations.
• Extension of OIR/PSF methods to three- and higher-point propagators with manageable overcompleteness.
• Robust schemes for analytic continuation tailored to OIR/PSF expansions.
• Characterization of the geometry and convergence landscape of PSF-manifolds in unmixing tasks, including adaptation to more general noise and baseline models (Dirnböck et al., 2024, Michelena et al., 24 Feb 2025).