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The distance between the second and first eigenvalues for the Dirichlet Laplacian of a domain is called its (spectral) gap. We show that the gap of a convex planar domain$D$symmetric about both the$x$and$y$axes is no smaller than the gap of an oriented rectangle which contains$D$.

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.

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