A kernel-derived orthogonal basis for spectral functions from Euclidean correlators

Abstract

Spectral functions play a central role in the characterization of a wide range of physical systems, including strongly interacting quantum field theories and many-body systems. Their non-perturbative determination from Euclidean correlation functions constitutes a well-known ill-posed inverse problem and has motivated the development of numerous reconstruction techniques. In this work, we propose a systematic and prior-free framework to represent spectral functions using an orthogonal functional basis derived directly from the kernel of Euclidean two-point correlation functions. By differentiating and partially integrating the Euclidean correlator over restricted time intervals, we identify a set of lattice-accessible constraints and the associated basis functions. These functions can be reorganized into an orthogonal basis with respect to which the spectral function may be approximated in a controlled manner. We demonstrate, using several model spectral functions, that the proposed expansion captures global features of the spectral function and reproduces low-energy transport coefficients with good accuracy when the spectral function is sufficiently smooth. While numerical implementation requires high-precision input for the Euclidean correlator, the present framework is intended as a tool to extract robust constraints and overall structures of spectral functions, rather than as a direct reconstruction method. The approach may thus serve as a complementary ingredient or preprocessing step for existing spectral reconstruction techniques.

Figures (4)

• Figure 1: The behaviors of the basis functions, $\tilde{S}_n^{(m)}(\tilde{k}^0)$ for $n=0, 1, 2, 3, 4$ and $m=1, 2, 3$. Different colors represent different $n$.
• Figure 2: The eigenvalue distribution for the different sizes of matrix $X$ which is $i\times i$.
• Figure 3: The models for the spectral function (solid curve) and the constraints obtained from it (dots) for the models $\tilde{\rho}_1$ (left), $\tilde{\rho}_2$ (middle) and $\tilde{\rho}_3$ (right).
• Figure 4: The approximated $\tilde{\rho}/\tilde{\omega}$ and the input (black curve) for the model 1 (top), 2 (middle) and 3 (bottom).