Formulation of the Results

For the sake of convenience we will denote independent variables by $x$ and $y$ and put $W(x,y)=V(x)+U(y)$. We begin with the operator $H_{a}$ on $L^{2}({\mathbb R}^{6})$, such that

H_{a} u = -\Delta_{x} u + V(x) u -\Delta_{y}u+U(y)u- a u

with the Laplace operators $\Delta_{x}$ and $\Delta_{y}$ in $x=(x_{1},x_{2}, x_{3})$ and $y=(y_{1},y_{2},y_{3})$, $a\geq 0$ is a parameter, the potentials $V(x)$ and $U(y)$ decay to zero as $x,
y \to \infty$. We investigate the conditions on the function $f(x,y)\in L^2 (\mathbb{R}^6)$ under which the equations
\end{displaymath} (22)

\end{displaymath} (23)

the second one is the limiting case of the first one as $a\to 0$, have the unique solution in $L^2(\mathbb{R}^6)$. Thus the case of a single Schrödinger operator studied in [#!VV08!#] is being generalized to the case of the sum of two such operators. We will use the spectral decomposition of self-adjoint operators.

For a function $\psi(x)$ belonging to a $L^{p}({\mathbb R}^{d})$ space with $1\leq p\leq \infty, \ d\in {\mathbb N}$ its norm is being denoted as $\Vert\psi\Vert _{L^{p}({\mathbb R}^{d})}$. As technical tools for estimating the appropriate norms of functions we will be using, in particular Young's inequality

\Vert f_{1}*f_{2}\Vert _{L^{\infty} (\mathbb{R}^3)}\leq
...\mathbb{R}^3), \quad
f_{2}\in L^{\frac{4}{3}} (\mathbb{R}^3),

where * stands for the convolution. The inner product of functions on $L^{2}({\mathbb R}^{d}), \ d\in {\mathbb N}$ is being denoted as

(f_{1}(x),f_{2}(x))_{L^{2}({\mathbb R}^{d})}:= \int_{{\mathbb
R}^{d}}f_{1}(x){\bar f_{2}}(x)dx ;

for a vector function $A(x)=(A_{1}(x), A_{2}(x), A_{3}(x))$, $x\in {\mathbb R}^{3}$ the inner product $(f_{1}(x), A(x))_{L^{2}({\mathbb R}^{3})}$ is the vector with the coordinates $\int_{{\mathbb R}^{3}} f_{1}(x){\bar
A_i}(x)dx$, $i=1,2,3$. Note that with a slight abuse we use the same notation even when functions may not be square integrable, for instance the functions $\varphi_{k}(x)$ and $\eta_{q}(y)$ of the continuous spectrum of the operators $-\Delta_{x}+V(x)$ and $-\Delta_{y}+U(y)$ respectively are normalized to Dirac delta-functions (see (3.1) and (3.2) in Section 3).

We make the following technical assumptions on the potential functions involved in the equations ([*]) and ([*]) and on the right sides of these equations.
{\sl The potential functions
$V(x), U(y):{\mathbb
...U\Vert _{L^{3\over 2}({\mathbb R}^{3})}< 4\pi
\end{displaymath} \end{assumption}
Here $c_{HLS}$ is the constant in the Hardy-Littlewood-Sobolev inequality

\Big\vert \int_{{\mathbb R}^{3}}\int_{{\mathbb R}^{3}} {f_{1...
...mathbb R}^{3})}^
{2}, \ f_{1}\in L^{3\over 2}({\mathbb R}^{3})
\end{displaymath} (24)

and given on p. 98 of [#!LL97!#]. The function

f(x,y)\in L^{2}({\mathbb R}^{6}) \quad \text{and} \quad \ver...
...thbb R}^{6}), \ \vert y\vert f(x,y)\in L^{1}({\mathbb R}^{6}).

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 2

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