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

Abstract

The non-real spectrum of a singular indefinite Sturm–Liouville operatorA=1r(−ddxpddx+q) with a sign changing weight function r consists (under suitable additional assumptions on the real coefficients 1/p,q,r∈Lloc1(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∈Ls(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.