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In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operator$D$(α)=$D$_{0}−α$V$1 acting in$L$_{2}($R$^{3};$C$^{4}) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.

A new proof is given for a theorem by V. G. Maz'ya. It gives a necessary and sufficient condition on the open set Ω in$R$^{$N$}for the functions in$W$_{0}^{$m,p$}(Ω) to have the ordinary norm equivalent to the norm obtained when including only the highest order derivatives in the definition. The proof is based on a kind of polynomial capacities, Maz'ya capacities.

A fundamental solution of the acoustical equation with a variable refraction coefficient is constructed. The solution satisfies the limiting absorption and radiation conditions. The optimal high frequency estimate is proved for square means of the solution. The source function for the diffusion equation is a by-product of this construction.