The paper deals with the eigenvalue problem \( -\Delta u - \alpha u + \lambda g(x) u = 0 \) with \(u \in H^1(\mathbb R^N)\), \(u\neq 0\), where \(\alpha, \lambda \in \mathbb R\), and some restrictions are imposed on the function \(g\). The function \(g\) represents a potential well that deepens as \(\lambda > 0\) increases. The main question is: For given \(\alpha > 0\), does there exist a value \(\lambda > 0\) for which the problem has a positive solution? It is shown that this occurs if and only if \(\alpha\) lies in a certain interval \((\Gamma, \xi_1)\) and that in this case the value of \(\lambda\) is unique, \(\lambda=\Lambda(\alpha)\). The properties of the function \(\Lambda(\alpha)\) are also discussed. Here \(\xi_1\) is the first eigenvalue of the Dirichlet problem \( -\Delta \phi = \xi \phi \) in \(\Omega\), \(\phi \in H^1_0(\Omega)\), and \(\Omega\) satisfies the same conditions as the function \(g\). It should be noted that this paper involves describing the eigenvalue \(\lambda\) as a function of the parameter \(\alpha\) rather than the eigenvalue \(\alpha\) as a function of the parameter \(\lambda\) in the traditional treatment.