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

$$H_{a}u=f$$ (2)

and

$$H_{0}u=f,$$ (3)

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

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

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.

Here $c_{HLS}$ is the constant in the Hardy-Littlewood-Sobolev inequality

$$\Big\vert \int_{{\mathbb R}^{3}}\int_{{\mathbb R}^{3}} {f_{1... ...mathbb R}^{3})}^ {2}, \ f_{1}\in L^{3\over 2}({\mathbb R}^{3})$$ (4)

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

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

writefiletoc writefiletoc writefiletoc