Approximation theory for Green's functions via the Lanczos algorithm

It is known that Green's functions can be expressed as continued fractions, the content at the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>n</a:mi><a:mi>th</a:mi></a:mrow></a:math> level of the fraction is encoded in a coefficient <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>b</b:mi><b:mi>n</b:mi></b:msub></b:math>, 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>N</c:mi></c:math> 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 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 coefficients. Typically, the decay of the error term ranges from <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mn>1</d:mn><d:mo>/</d:mo><d:mi>poly</d:mi><d:mo>(</d:mo><d:mi>N</d:mi><d:mo>)</d:mo></d:mrow></d:math> in the best case to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mn>1</e:mn><e:mo>/</e:mo><e:mi>poly</e:mi><e:mo>(</e:mo><e:mo form="prefix">log</e:mo><e:mi>N</e:mi><e:mo>)</e:mo></e:mrow></e:math> 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 behaviors of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>b</g:mi><g:mi>n</g:mi></g:msub></g:math>, and we also conjecture a relationship between the asymptotic behavior of the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>b</h:mi><h:mi>n</h:mi></h:msub></h:math>'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.