The non-real spectrum of a singular indefinite Sturm-Liouville operatorwith a sign changing weight function r consists (under suitable additional assumptions on the real coefficients 1/p, q, r ∈ L 1 loc (R)) of isolated eigenvalues with finite algebraic multiplicity which are symmetric with respect to the real line. In this paper bounds on the absolute values and the imaginary parts of the non-real eigenvalues of A are proved for uniformly locally integrable potentials q and potentials q ∈ L s (R) for some s ∈ [1, ∞]. The bounds depend on the negative part of q, on the norm of 1/p and in an implicit way on the sign changes and zeros of the weight function.
The spectrum of the singular indefinite Sturm-Liouville operator A = sgn(·) − d 2 dx 2 + q with a real potential q ∈ L 1 (R) covers the whole real line and, in addition, non-real eigenvalues may appear if the potential q assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction were obtained. In this paper the bound |λ| ≤ q 2 L 1 on the absolute values of the non-real eigenvalues λ of A is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the L 1 -norm of the negative part of q.
A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(·) and an integrable, continuous potential q is obtained.
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