The paper is devoted to studying the spectral properties of the selfadjoint operator \({\mathcal H}\) (the so-called Hamiltonian of the Stark effect) generated by the differential expression \[ Hu= -d^2u/dx^2+ xu+ q(x)u \] with real-valued coefficient \(q(x)\) under the conditions \[ \Omega(a)= \int^\infty_{-\infty} |q(x)+ a|(|x|+1)^{-{1\over 2}}dx<+\infty\tag{1} \] for any real \(a\), and \[ \Omega(a)\to 0\quad\text{as }a\to\pm\infty.\tag{2} \] Firstly, the author performs the spectral analysis of the simplest Hamiltonian \({\mathcal H}_0\) corresponding to the case \(q(x)\equiv 0\). Namely, the spectrum of \({\mathcal H}_0\) fills the entire real axis \(\mathbb{R}\) and is absolutely continuous, and \({\mathcal H}_0\) admits an ordered spectral representation of the space \(L_2(\mathbb{R})\) which has multiplicity one and is characterized by the spectral measure \(\rho_0(\lambda)=\lambda\).
Further, the generalized eigenfunctions \(a(x,\lambda)\) of \({\mathcal H}\) are constructed as solutions to some integral equation. Finally, a representation of the resolvent kernel of \({\mathcal H}\) is obtained on the basis of developed estimates of the eigenfunctions \(a(x,\lambda)\).