For the sake of convenience we will denote independent variables
by and and put
. We begin with the
, such that
with the Laplace operators and in
a parameter, the potentials and decay to zero as
. We investigate the conditions on the function
under which the equations
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
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
is being denoted
for a vector function
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
the continuous spectrum of the operators
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
and given on p. 98 of [#!LL97!#].