Geodesic
On the convergence in $H^{s}$-norm of the spectral expansions corresponding to the differential operators with singularity
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 1125-1128 
V. S. Serov. On the convergence in $H^{s}$-norm of the spectral expansions corresponding to the differential operators with singularity. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 1125-1128. http://geodesic.mathdoc.fr/item/FPM_1995_1_4_a24/
@article{FPM_1995_1_4_a24,
     author = {V. S. Serov},
     title = {On the convergence in $H^{s}$-norm of the spectral expansions corresponding to the differential operators with singularity},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1125--1128},
     year = {1995},
     volume = {1},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1995_1_4_a24/}
}
TY  - JOUR
AU  - V. S. Serov
TI  - On the convergence in $H^{s}$-norm of the spectral expansions corresponding to the differential operators with singularity
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 1995
SP  - 1125
EP  - 1128
VL  - 1
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/FPM_1995_1_4_a24/
LA  - ru
ID  - FPM_1995_1_4_a24
ER  - 
%0 Journal Article
%A V. S. Serov
%T On the convergence in $H^{s}$-norm of the spectral expansions corresponding to the differential operators with singularity
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 1995
%P 1125-1128
%V 1
%N 4
%U http://geodesic.mathdoc.fr/item/FPM_1995_1_4_a24/
%G ru
%F FPM_1995_1_4_a24

Voir la notice de l'article provenant de la source Math-Net.Ru

In this work we prove the convergence in the norm of the Sobolev spaces $H^s(\mathbb R^{N})$ of the spectral expansions corresponding to the self-adjont extansions in $L^2(\mathbb R^{N})$ of the operators in the following way: $$ A(x,D)=P(D)+Q(x), $$ where $P(D)$ is the self-adjont elliptic operator with constant coefficients and of order $m$ and real potential $Q(x)$ belongs to Kato space. As a consequence of this result we have the uniform convergence of these expansions for the case $m>\frac{N}{2}$.

[1] Schechter M., Spectra of partial differential operators, North-Holland, Amsterdam, London, 1971, 268 pp. | MR | Zbl

[2] Alimov Š. A., Joó I., “On convergence of eigenfunction expansions in $H^s$-norm”, Acta Sci. Math., 48 (1985), 5–12 | MR | Zbl

[3] Alimov Š. A., Barnovska M., “On eigenfunction expansions connected with the Schrödinger operator”, Slovak. Math. J., 1985

[4] Ilin V. A., “O skhodimosti razlozhenii po sobstvennym funktsiyam operatora Laplasa”, UMN, 13:1 (1958), 87–180 | MR

[5] Ilin V. A., Spektralnaya teoriya differentsialnykh operatorov, Nauka, M., 1991 | MR