**Fredholm type solvability conditions,
which affirm that an operator equation is solvable if and only if
its right-hand side is orthogonal to solutions of the homogeneous adjoint equation,
are used directly or indirectly in the most methods of linear and nonlinear
analysis.
If the operator does not satisfy the Fredholm property,
applicability of these solvability conditions is not established.
In this work we study solvability conditions for some class of
non Fredholm operators. We consider the equation
**

**There are two distinct cases, and . Let us first
discuss the case . It is known that homogeneous elliptic
operators with constant coefficients satisfy the Fredholm
property if considered in some specially chosen polynomial weighted
spaces.
For the operator given by () this is the case
if and , that is for the Laplace operator.
Lower order terms with the coefficients decaying at infinity
represent compact operators if the decay rate is sufficiently high.
Therefore the operator in the weighted
spaces remains Fredholm under some conditions on the potential.
This allows one to make some conclusions about its index and
solvability conditions.
This approach is based on a priori estimates of solutions obtained
in [#!Nirenberg-Walker1973!#].
The Fredholm property of this class of operators in Sobolev spaces is studied in
[#!Lokhart1981!#], [#!Lokhart-McOwen1983!#].
Similar problems for elliptic operators in Hölder spaces are
investigated in [#!Benkirane1988!#], [#!Bolley1993!#].
Exterior problems for the Laplace operator in weighted Sobolev
spaces are considered in [#!Amrouche1997!#],
[#!Amrouche2008!#], and for more general operators in Hölder spaces in
[#!Bolley2001!#].
The dimension of the kernel and the Fredholm property of elliptic operators of the first
order are studied in [#!Walker1971!#], [#!Walker1972!#].
**

**The case of positive is qualitatively different. The method
described above is not applicable. In the 1D case we can
introduce exponential weighted spaces where the operator will
satisfy the Fredholm property [#!VV06!#]. However, in with
this method is not applicable neither.
The reason for this can be already seen for the case
. If the equation is solvable, then the right-hand side is
orthogonal to all functions
, where
,
.
Hence there is a continuous family of solvability conditions
while the Fredholm property implies only a finite number of them.
**

**The method developed in our previous work is based on the
spectral theory of self-adjoint operators [#!VV08!#]. Similar to the case
where we can use the Fourier transform and explicitly find the
solution, in the case of nonzero potential we use spectral
decomposition with respect to the functions of the continuous
spectrum of the operator
.
This allows us to obtain solvability conditions as orthogonality
to the functions of the continuous spectrum.
To the best of our knowledge, this is the first result on
solvability conditions for this class of operators.
This method is applicable both for and .
Though these solvability conditions
are similar to the usual ones, we should not forget that the
operator does not satisfy the Fredholm property. Its range is not
closed, the dimension of the kernel and the codimension of the
image may not be finite. Hence, the similarity with Fredholm
solvability conditions is only formal.
**

**In this work we continue the investigation of equation
() under different assumptions on the potential.
We will assume that it can be represented as
, where
,
,
.
Though the potential does not converge to zero as
, we can apply the method of [#!VV08!#] using separation of
variables.
A particular case of such equations with
arises in reaction-diffusion problems [#!VKMP02!#].
**