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SUMMARY:Eigenvalues of non self-adjoint Toeplitz operators
DTSTART:20260504T120000Z
DTEND:20260504T130000Z
DTSTAMP:20260501T211700Z
UID:indico-event-15310@indico.math.cnrs.fr
DESCRIPTION:Speakers: Nathan Réguer (IRMAR - Université de Rennes)\n\nWe
  are interested in semiclassical operators built from quantisation of symb
 ols $Op_{\\hbar}(f)$\, in particular on their eigenvalues. Actually\, eige
 nvalues are the poles of the resolvent\, and for self-adjoint operators\, 
 the norm of the resolvent at a number $E$ is equal to the inverse of the d
 istance from $E$ to the spectrum. Hence\, finding quasimodes\, in other wo
 rds eigenfunctions up to a small remainder with respect to $\\hbar$: $Op_{
 \\hbar}(f)u_{\\hbar} = E u_{\\hbar} + O(\\hbar^{\\infty})$\, gives eigenva
 lues up to a small remainder too. Although\, it is no longer true for non 
 self-adjoint operators. Fortunately\, it is still possible to get informat
 ion from exponentially close quasimodes $Op_{\\hbar}(f)u_{\\hbar} = E u_{\
 \hbar} + O(e^{-\\frac{c}{\\hbar}})$.First I will present my problem\, whic
 h is to find the shape of the spectrum near a point $f(x_0)$ such that $df
 (x_0)=0$ and the Hessian of $f$ at $x_0$ satisfies an ellipticity conditio
 n. Then I will present the framework I use for this study: Toeplitz operat
 ors\, which is convenient as the state functions are defined directly on t
 he phase space. For these operators\, we will see the specific case of qua
 dratic symbols\, for which the spectrum is well-known since a long time\, 
 but the main ingredient for our method will appear in the proof. To treat 
 the general symbol\, we will consider Fourier integral operators with comp
 lex phase\, and symbolic calculus on Toeplitz operators\, with the specifi
 city that all the data have to be analytic. In a last part\, I will give s
 ome elements of the proof of the main result.\n\nhttps://indico.math.cnrs.
 fr/event/15310/
LOCATION:Salle Pellos
URL:https://indico.math.cnrs.fr/event/15310/
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