For the sake of convenience we will denote independent variables
by and and put
. We begin with the
operator on
, such that

with the Laplace operators and in and , is a parameter, the potentials and decay to zero as . We investigate the conditions on the function under which the equations

and

the second one is the limiting case of the first one as , have the unique solution in . 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 belonging to a
space with
its norm is
being denoted as
. As technical
tools for estimating the appropriate norms of functions we will be
using, in particular Young's inequality

where * stands for the convolution. The inner product of functions on is being denoted as

for a vector function , the inner product is the vector with the coordinates , . Note that with a slight abuse we use the same notation even when functions may not be square integrable, for instance the functions and of the continuous spectrum of the operators and 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 is the constant in the Hardy-Littlewood-Sobolev inequality

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