Spectral bounds for indefinite singular Sturm–Liouville operators with uniformly locally integrable potentials

Abstract

Abstract The non-real spectrum of a singular indefinite Sturm–Liouville operator A = 1 r ( − d d x p d d x + q ) with a sign changing weight function r consists (under suitable additional assumptions on the real coefficients 1 / p , q , r ∈ L loc 1 ( 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.