Consider the regular indefinite Sturm-Liouville eigenvalue problem \[ \tau(f) := ( -(pf')' + q f )/w = \lambda f \] on a bounded interval \((a,b)\) with real coefficients \(w, q, 1/p \in L^1(a,b)\) satisfying \(p(x) > 0, \, w(x) \neq 0\) a.e.. Assuming that the weight function \(w\) changes its sign, the inner product \([f,g] := \int f \overline{g} \, w \; dx\) defines a Krein space structure on \(L^2_{|w|}(a,b)\). It is well known that the spectrum of each self-adjoint operator realization \(A\) of \(\tau\) in this Krein space consists of real eigenvalues accumulating at \(+ \infty\) and at \(- \infty\) and of a finite number of non-real eigenvalues which are placed symmetrically with respect to the real axis.
Improving some earlier results for particular cases (e.g. [B. Xie and J. Qi, J. Differ. Equations 255, No. 8, 2291–2301 (2013; Zbl 1291.34052); the second and the fourth author, J. Spectr. Theory 4, No. 1, 53–63 (2014; Zbl 1302.34128)]) the present paper provides estimates on bounds for the real and imaginary parts of the non-real eigenvalues. These bounds involve the self-adjoint boundary conditions, the coefficients \(p\), \(q\) and, in a somehow weak form, also the weight \(w\). For the case of finitely many sign changes of \(w\) this result is formulated more explicitly. Furthermore, the result is improved for the case \(\int w \; dx \neq 0\). Finally, also estimates on bounds for the so-called exceptional real eigenvalues are obtained. These are real eigenvalues such that the form \([A \cdot, \cdot]\) is not positive definite on the associated root subspace.