On the spectrum of a class of differential operators and some imbedding theorems

The spectral properties of the operator
$$ Lu=\frac{(-1)^{n+k}}{\rho(t)}\biggl(\frac{u(t)}{\rho(t)}\biggr)^{(2n+2k)}+\frac{(-1)^k}{\rho(t)}\biggl(V^2(t)\biggl(\frac{u(t)}{\rho(t)}\biggr)^{\!(k)}\biggr)^{\!(k)}\qquad(n,k\geqslant1) $$
are considered. A criterion for the nuclearity of the resolvent, two-sided estimates of the nuclear norm, Green's functions on the diagonal and the norms of the inverse operator are obtained.
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