Estimate of fourier transforms with respect to the system of generalized eigenfunctions of the Schrödinger operator with Stummel-type potential

Abstract

Suppose thatА is a nonnegative self-adjoint extension to {$\mathbb{R}^{\rm N} \left( {N \geqslant 1} \right)$} of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {

$${}_{x \in R^N }^{\sup } \int {_{\left| {x - y} \right| \leqslant 1} ^{\left| {q\left( y \right)} \right|^2 dy< \infty {\text{ }}N = 1,2,3,} } $$

} or the condition {

$${}_{x \in R^N }^{\sup } \int {_{\left| {x - y} \right| \leqslant 1} ^{\left| {q\left( y \right)} \right|^{4 - N} \chi \left( {\left| {x - y} \right|} \right)\left| {{\text{q}}\left( {\text{y}} \right)} \right|^{\text{2}} {\text{dy< }}\infty {\text{ }}N \leqslant 4,} } $$

} in which the nonnegative function itχ(r) is such that {$\smallint _0 ^1 \left( {\chi \left( r \right)r} \right)^{ - 1} dr< \infty$}. For each α∈(0, 2], we establish an estimate of the generalized Fourier transforms of an arbitrary function {$\smallint \in L{}_2^a (\mathbb{R}^N )$} of the form {

$$\sum\limits_{i = 1}^m {\int_0^\infty {\left| {\hat f_i \left( \leftthreetimes \right)} \right|^2 \left( {1 + \leftthreetimes } \right)^a d\rho \left( \leftthreetimes \right) \leqslant M\left\| f \right\|^2 _{L_2^a \left( {R^2 } \right)} } } $$

} If, in addition, {$\lim _{r \to 0 + 0} x\left( r \right) = + \infty$}, then, along with this estimate, a similar lower bound is established.