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Abstract

Let A be a pseudodifferential operator on $ℝ^N$ whose Weyl symbol a is a strictly positive smooth function on $W = ℝ^N × ℝ^N$ such that $|∂^{α}a| ≤ C_αa^{1-ϱ}$ for some ϱ>0 and all |α|>0, $∂^{α}a$ is bounded for large |α|, and $lim_{w→∞}a(w) = ∞$. Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.