Consider the following Sturm-Liouville eigenvalue problem \[ -u''(x)+ q(x) u(x)= \lambda u(x) \tag{1} \] \[ u'(0)= 0,\;u'(\pi)- m\lambda u(\pi) =0 \tag{2} \] where \(m\) is a physical parameter and \(\lambda\) is a spectral parameter, appearing also in the second boundary condition. The author studies the asymptotic behavior of eigenvalues and eigenfunctions of the problem as \(m\) tends to zero. The author presents the spectral problem as an eigenvalue problem for a linear operator pencil in the Hilbert space \(H=L_2 \oplus C\).