Hyperbolic equation method in \(L^ p\)-spaces (Q909879)

The hyperbolic equation method is proposed by A. Povsner (1953) and developed in the works of Yu. Berezanskij, Yu. Orochko, P. Chernoff, T. Kato, etc. This is an effective method for studying essential selfadjointness of the operator \[ H=-\sum^{\ell}_{k,j=1}(\partial /\partial x_ k)a_{kj}\partial /\partial x_ j\quad in\quad L^ 2({\mathbb{R}}^{\ell},d^{\ell}x) \] on the set of finite functions. Using ideas of \textit{Yu. Orochko} [Rep. Math. Phys. 15, 163-172 (1979; Zbl 0415.35022)], a generalization of the method in the Banach space \(L^ p({\mathbb{R}}^{\ell},d^{\ell}x)\) is proposed.