Approximation theory for Green's functions via the Lanczos algorithm

• URL: http://arxiv.org/abs/2505.00089v1
• Date: Wed, 30 Apr 2025 18:00:43 GMT
• Title: Approximation theory for Green's functions via the Lanczos algorithm
• Authors: Gabriele Pinna, Oliver Lunt, Curt von Keyserlingk,
• Abstract summary: It is known that Green's functions can be expressed as continued fractions.<n>We present a theory concerning errors in approximating Green's functions using continued fractions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: It is known that Green's functions can be expressed as continued fractions; the content at the $n$-th level of the fraction is encoded in a coefficient $b_n$, which can be recursively obtained using the Lanczos algorithm. We present a theory concerning errors in approximating Green's functions using continued fractions when only the first $N$ coefficients are known exactly. Our focus lies on the stitching approximation (also known as the recursion method), wherein truncated continued fractions are completed with a sequence of coefficients for which exact solutions are available. We assume a now standard conjecture about the growth of the Lanczos coefficients in chaotic many-body systems, and that the stitching approximation converges to the correct answer. Given these assumptions, we show that the rate of convergence of the stitching approximation to a Green's function depends strongly on the decay of staggered subleading terms in the Lanczos cofficients. Typically, the decay of the error term ranges from $1/\mathrm{poly}(N)$ in the best case to $1/\mathrm{poly}(\log N)$ in the worst case, depending on the differentiability of the spectral function at the origin. We present different variants of this error estimate for different asymptotic behaviours of the $b_n$, and we also conjecture a relationship between the asymptotic behavior of the $b_n$'s and the smoothness of the Green's function. Lastly, with the above assumptions, we prove a formula linking the spectral function's value at the origin to a product of continued fraction coefficients, which we then apply to estimate the diffusion constant in the mixed field Ising model.

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