Carleman estimates for the Schr\"odinger operator with a~locally semibounded strongly singular potential

Yu. B. Orochko

Sbornik. Mathematics, Tome 33 (1977) no. 1, pp. 147-158

Résumé

Let $A$ be an arbitrary selfadjoint extension in $L_2(\mathbf R^n)$, $n\geqslant2$, of the minimal Schrödinger operator with a potential $q(x)\in L_{2,\mathrm{loc}}(\mathbf R^n)$ that is locally bounded from below. For a certain class of functions $\Phi(A,t)$ of $A$ and a parameter $t>0$, which are connected with the hyperbolic equation $u''=Au$, an estimate of the form $$ \bigl|[\Phi(A,t)f](x)\bigr|\leqslant c(x,t)\int_{|y-x|\leqslant t}|f(y)|^2\,dy $$ is obtained for almost all $x\in\mathbf R^n$; here $f\in L_2(\mathbf R)^n$ is a function with compact supportand $c(x,t)$ is explicitly expressed in terms of an arbitrary continuous function $m(x)\geqslant-q(x)$, $x\in\mathbf R^n$. An application of this estimate to the question of pointwise approximation of functions by spectral “wave packets” is considered. Bibliography: 15 titles.