Sufficient conditions that the minimum and maximum of partial differential operators should coincide and that their spectra should be discrete

M. G. Gimadislamov

Matematičeskie zametki, Tome 4 (1968) no. 3, pp. 301-312.

Résumé

An expression of the form $$ l(u)=(-1)^m\sum_{j=1}^m D_j^{2m}u+[q(x)+ir(x)]u $$ is considered. Sufficient conditions are found such that the minimum operator, formally conjugate to $l(u)$, generated by the expression and the maximum operator generated by the expression $l(u)$ in $\mathscr{L}_2(E_n)$ should coincide. It is proved that if $q(x)\to\infty$ or $q(x)+r(x)\to\infty$, $|x|\to\infty$, then the operator generated by $l(u)$ in $\mathscr{L}_2(E_n)$ has a discrete spectrum.