bravchick (bravchick) wrote in geometry_qa,
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bravchick
geometry_qa

Reference for variational formula for the determinant of an elliptic operator

Let A be a (note necessarily self-adjoint) invertible elliptic operator. The following variational formula for its determinant is well known

δ Det(A) :=  δ (-∂s Tr ( A-s ) |s=0)  = s  ( s Tr[(δ A ) A-s-1])|s=0

cf., for example, Burghelea, Friedlander, Kappeler, Meyer-Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107 (1992), 34--65, or Kontsevich, Vishik , Geometry of determinants of elliptic operators. (of course, to define A-s one uses a spectral cut, which I did not write explicitly to simplify the notation).

Is it proven anywhere in the literature? (I do know how to prove it, but I need a reference)
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