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Mathematics of Computation

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On the evaluation of Matsubara sums

Author: Olivier Espinosa
Journal: Math. Comp. 79 (2010), 1709-1725
MSC (2000): Primary 33-XX; Secondary 33E20, 33F99
Published electronically: November 19, 2009
MathSciNet review: 2630009
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Abstract: Given a connected (multi)graph $ G$, consisting of $ V$ vertices and $ I$ lines, we consider a class of multidimensional sums of the general form

$\displaystyle S_G:= \sum\limits_{n_1 = - \infty }^\infty \sum\limits_{n_2 = - \... \left( {n_2^2 + q_2^2 } \right) \cdots \left( {n_I^2 + q_I^2 } \right)}}}, $

where the variables $ q_i$ ( $ i=1,\ldots,I$) are real and positive and the variables $ N_v$ ( $ v=1,\ldots,V$) are integer-valued. $ \delta_G(n_1 ,n_2 , \ldots ,n_I;\{N_v\} )$ is a function valued in $ \{0,1\}$ which imposes a series of linear constraints among the summation variables $ n_i$, determined by the topology of the graph $ G$.

We prove that these sums, which we call Matsubara sums, can be explicitly evaluated by applying a $ G$-dependent linear operator $ \hat{\mathcal{O}}'_G$ to the evaluation of the integral obtained from $ S_G$ by replacing the discrete variables $ n_i$ by continuous real variables $ x_i$ and replacing the sums by integrals.

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Additional Information

Olivier Espinosa
Affiliation: Departamento de Física, Universidad Téc. Federico Santa María, Valparaíso, Chile

Received by editor(s): March 13, 2009
Received by editor(s) in revised form: May 19, 2009
Published electronically: November 19, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.