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# Conférence - DimaScat : Théorie du scattering et asymptotiques spectrales d'opérateurs différentiels - en l'honneur de Dimitri Yafaev

Mercredi, 20 Avril, 2016 - 09:30 - 10:30
Let $U$, $V$ be a Weyl pair $UV=q^2VU$ of unbounded self-adjoint operators, realized in $L^2(\mathbb{R})$ as $U=e^{\alpha Q}$ and $V=e^{\beta P}$, where $P$, $Q$ are canonical operators and $\alpha$, $\beta$ are positive real numbers. The operator $L=U+U^{-1}+V$ plays an important role in the theory of quantum group $SL_q(2,\mathbb{R})$, cluster algebras, quantum Liouville model, quantum Teichmuller theory. We show that $L$ has simple continuous spectrum in the interval $[2,+\infty]$. We construct its resolvent and a complete family of eigenfunctions. The talk is based on a joint article with L. Takhtajan.