We study spectral and scattering properties of the Laplacian$H$^{(σ)}=-Δ in $L_2(\mathbf{R}^{d+1}_+)$ corresponding to the boundary condition $\frac{\partial u}{\partial\nu} + \sigma u = 0$ with a periodic function σ. For non-negative σ we prove that$H$^{(σ)}is unitarily equivalent to the Neumann Laplacian$H$^{(0)}. In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of$H$^{(σ)}under mild assumptions on σ.

No abstract uploaded!

Let $\mathcal{D}$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function$f$∈ $\mathcal{D}$ to be$cyclic$, i.e. for {$pf$:$p$is a polynomial} to be dense in $\mathcal{D}$ .

We find a real analytic Levi-flat hypersurface in$C$^{2}containing a bounded contractible domain which is a determining set for pluriharmonic functions.

We study the problem of density of polynomials in the de Branges spaces ℋ($E$) of entire functions and obtain conditions (in terms of the distribution of the zeros of the generating function$E$) ensuring that the polynomials belong to the space ℋ($E$) or are dense in this space. We discuss the relation of these results with the recent paper of V. P. Havin and J. Mashreghi on majorants for the shift-coinvariant subspaces. Also, it is shown that the density of polynomials implies the hypercyclicity of translation operators in ℋ($E$).