 Solution

For a separated harmonic function $V=F(x)G(y)$, Laplace's equation gives $F^{\prime \prime }/F=-G^{\prime \prime }/G=-k^2$. Bounded periodic dependence in $x$ has $F=A\cos (kx)+B\sin (kx)$; decay on each half-plane selects $G=e^{-k|y|}$ for $k>0$, when the potential is matched continuously at the sheet. These are the separated decaying modes. General decaying solutions can be superpositions of them; decay alone does not force a single value of $k$.

The prescribed cosine density selects $k=1$ and $V=C\cos x{e}^{-|y|}$. Its normal derivative jump is $-2C\cos x$. Matching the stated jump $2\pi {\Sigma }_0\cos x$ givesThis is the gravitational potential of a sinusoidal sheet in the normalization of the question. It is continuous at the sheet and decays on both sides. Since ${\Sigma }_0\cos x$ changes sign, it can physically describe a Fourier component of a density perturbation rather than an everywhere positive isolated mass sheet.