We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle,$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$where $\Pi$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating...

We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators $T_g$ on $H^p\left(D\right)$, where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of $T_g$ provided that $g^{-1}\left(0\right)$ is a compact subset of D.

Let $\mu$ be a positive Borel measure on the complex plane ${ℂ}^n$ and let $j=(j_1,\cdots ,j_n)$ with $j_i\in \mathbb{N}$. We study the generalized Toeplitz operators $T_{\mu }^{\left(j\right)}$ on the Fock space $F_{\alpha }^2$. We prove that $T_{\mu }^{\left(j\right)}$ is bounded (or compact) on $F_{\alpha }^2$ if and only if $\mu$ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class for $1\le p<\infty$.

In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy $E$. This permits a detailed study of the spectrum in various asymptotic regions of the parameters $(E,\hslash )$, and gives improvements and new proofs for many of the results in the field. In the completely resonant...

We consider the solution operator $S{ℱ}_{\mu ,(p,q)}\to L^2{\left(\mu \right)}_{(p,q)}$ to the $\overline{\partial }$-operator restricted to forms with coefficients in ${ℱ}_{\mu }=\left\{ff\text{is}\text{entire}\text{and}{\int }_{{ℂ}^n}{\left|f\left(z\right)\right|}^2\mathrm{d}\mu \left(z\right)<\infty \right\}$. Here ${ℱ}_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in ${ℱ}_{\mu }$, $L^2\left(\mu \right)$ is the corresponding $L^2$-space and $\mu$ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\overline{\partial }$. This solution operator will have the property $Sv\perp {ℱ}_{(p,q)}\forall v\in {ℱ}_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators...

Let $u$ be an inner function and $K_u^2$ be the corresponding model space. For an inner function $v$, the subspace $v{H}^2$ is an invariant subspace of the unilateral shift operator on $H^2$. In this article, using the structure of a Toeplitz kernel $\ker T_{\overline{u}v}$, we study the intersection $K_u^2\cap v{H}^2$ by properties of inner functions $u$ and $v$$(v...$