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.
UDC:
517.43
MSC: 35J10,
35B45 Received: 22.10.1976
Fulltext:
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