On the convergence in $H^{s}$-norm of the spectral expansions corresponding to the differential operators with singularity

V. S. Serov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 1125-1128

Résumé

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 $m>\frac{N}{2}$.