arXiv subjects
Publications and source records attributed to Mauricio D. Garay.
I prove, under mild assumptions, that solutions to linear evolution equations admit sectorial solutions. The size of the sector depends on the regularity of the initial data. If it is regular enough the solution is holomorphic and unique otherwise it is sectorial. I also prove that the result is optimal for many partial differential systems (which includes KdV and other examples).
We give a sufficient condition for quantising integrable systems.
Expository paper on the relations between perturbation theory of pseudo-differential operators, finiteness theorems and deformations of Lagrangian varieties.
We prove a D=1 analytic versal deformation theorem for WKB expansions. We define the spectrum of an operator in local analytic terms. We use the Morse lemma to show that the perturbation series arising in a perturbed harmonic oscillator become analytic after a formal Borel transform.
We use Kiehl-Verdier's and Houzel's finiteness theorems in the setting of local analytic geometry, and the Whitney-Thom theory of stratified spaces, to prove that fibrewise constructible complex of sheaves have coherent direct images. We review the notions of stratification theory which are needed.
We study the monodromy of vanishing cycles for map-germs $f:(C^{2n},0) \to (\CM^k,0)$ whose components are in involution. Although the singular fibres of such maps have non-isolated singularities, it is shown that the regular fibres are $2(n-k)$-connected and that the vanishing homology group of rank $2(n-k)+1$ is freely generated by the vanishing cycles. As corollaries, we get that the multiplicity of the discriminant is equal to the dimension of the vanishing homology group of rank $2(n-k)+1$ and that the Variation operator is an isomorphism. These results are proved under two assumptions: 1. the pyramidality assumptions which states that the singular locus is propagated along the Hamilton flow of the components of $f$ 2. the generic singular fibres should have transverse Morse singularities and their locus should be connected. It is conjectured that outside a set of infinite codimension the first condition holds and that there exists an involutive deformation of $f$ which satisfies condition 2.
We consider deformations of singular Lagrangian varieties in symplectic spaces. We show the coherence of the direct image sheaves of relative infinitesimal Lagrangian deformations. Using this result, we prove that, under some assumptions, a Lagrangian deformation is analytically rigid provided that this is the case infinitesimally. This gives generalisations of recent results by Colin de Verdière concerning Lagrangian curves.