An asymptotic formula for the density of states of the polyharmonic periodic operator (−δ)^{$l$}+$V$in$R$^{$n$},$n$≥2,$l$>1/2 is obtained. Special consideration is given to the case of the Schrödinger equation$n$=3,$l$=1,$V$being a periodic potential, where the second term of the asymptotic is found.

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.

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