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It has been shown that solutions of a number of many-body problems out ofequilibrium can be expressed in terms of Toeplitz determinants withFisher-Hartwig (FH) singularities. In the present paper, such Toeplitzdeterminants are studied numerically. Results of our numerical calculationsfully agree with the FH conjecture in an extended form that includes asummation over all FH representations (corresponding to different branches ofthe logarithms). As specific applications, we consider problems of Fermi edgesingularity and tunneling spectroscopy of Luttinger liquid with multiple-stepenergy distribution functions, including the case of population inversion. Inthe energy representation, a sum over FH branches produces power-lawsingularities at multiple edges.

Luttinger liquids with multiple Fermi edges: Generalized Fisher-Hartwig conjecture and numerical analysis of Toeplitz determinants