2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62749Consider 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.Spectral TheoryFunctional Analysis31C10; 35P05; 37J45; 37K05; 47J20; 58J52text