Determinants of pseudodifferential operators and complex deformations of phase space

Details Determinants of pseudodifferential operators and complex deformations of phase space Determinants of pseudodifferential operators and complex deformations of phase space

2001-11-28

arxiv.org/abs/math/0111292v1

Mathematics

Spectral Theory

Scientific paper

Consider an h-pseudodifferential operator P, whose symbol extends holomorphically to a tubular neighborhood of the real phase space and converges sufficiently fast to 1, so that the determinant of P is well-defined. We show that the modulus of this determinant is asymptotically bounded by an exponential of the integral of the logarithm of the modulus of the symbol along a certain complex deformation of the real phase space. Since there are many possible such deformations, we get a variational problem. The paper is devoted to the corresponding variational calculus.

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