Matematičeskie zametki, Tome 4 (1968) no. 3, pp. 301-312
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M. G. Gimadislamov. Sufficient conditions that the minimum and maximum of partial differential operators should coincide and that their spectra should be discrete. Matematičeskie zametki, Tome 4 (1968) no. 3, pp. 301-312. http://geodesic.mathdoc.fr/item/MZM_1968_4_3_a5/
@article{MZM_1968_4_3_a5,
author = {M. G. Gimadislamov},
title = {Sufficient conditions that the minimum and maximum of partial differential operators should coincide and that their spectra should be discrete},
journal = {Matemati\v{c}eskie zametki},
pages = {301--312},
year = {1968},
volume = {4},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_1968_4_3_a5/}
}
TY - JOUR
AU - M. G. Gimadislamov
TI - Sufficient conditions that the minimum and maximum of partial differential operators should coincide and that their spectra should be discrete
JO - Matematičeskie zametki
PY - 1968
SP - 301
EP - 312
VL - 4
IS - 3
UR - http://geodesic.mathdoc.fr/item/MZM_1968_4_3_a5/
LA - ru
ID - MZM_1968_4_3_a5
ER -
%0 Journal Article
%A M. G. Gimadislamov
%T Sufficient conditions that the minimum and maximum of partial differential operators should coincide and that their spectra should be discrete
%J Matematičeskie zametki
%D 1968
%P 301-312
%V 4
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_1968_4_3_a5/
%G ru
%F MZM_1968_4_3_a5
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.
