Introduction

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

$$H u \equiv -\Delta u + W(x) u - a u = f$$ (1)

with a non-negative constant $a$ and with some conditions on the potential $W(x)$ which will be specified below. If $W(x) \to 0$ as $\vert x\vert \to \infty$, the case studied in our previous work [#!VV08!#], then the essential spectrum of the operator $H$ consists of the half-axis $[-a,\infty)$ [#!M1!#], [#!M2!#], [#!VV06!#], [#!JMST!#]. Since it contains the origin, the operator does not satisfy the Fredholm property, and the usual solvability conditions for equation () are not applicable.

There are two distinct cases, $a=0$ and $a>0$. Let us first discuss the case $a=0$. 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 $H$ given by () this is the case if $a=0$ and $W(x) \equiv 0$, 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 $H$ 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 $a$ 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 $\mathbb R^n$ with $n \geq 2$ this method is not applicable neither. The reason for this can be already seen for the case $W(x) \equiv 0$. If the equation is solvable, then the right-hand side is orthogonal to all functions $e^{ipx_{1}}e^{iqx_{2}}$, where $p^2 + q^2 = a$, $x=(x_1,x_2) \in \mathbb R^2$. 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 $W(x) \equiv 0$ 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 $H_0 u = -\Delta u + W u$. 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 $a=0$ and $a>0$. 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 $W(x) = W_1(x') + W_2(x'')$, where $x=(x_1,...,x_n)$, $x'=(x_1,...,x_k)$, $x''=(x_{k+1},...,x_n)$. Though the potential does not converge to zero as $\vert x\vert \to \infty$, we can apply the method of [#!VV08!#] using separation of variables. A particular case of such equations with $W_1(x') \equiv 0$ arises in reaction-diffusion problems [#!VKMP02!#].