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Carleman estimates for the Schrödinger operator with a locally semibounded strongly singular potential
Yu. B. Orochko
Abstract:
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.
Received: 22.10.1976
Citation:
Yu. B. Orochko, “Carleman estimates for the Schrödinger operator with a locally semibounded strongly singular potential”, Math. USSR-Sb., 33:1 (1977), 147–158
Linking options:
https://www.mathnet.ru/eng/sm2942https://doi.org/10.1070/SM1977v033n01ABEH002418 https://www.mathnet.ru/eng/sm/v146/i1/p162
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