J. Math. Anal. Appl. 291 (2004) 246–261 www.elsevier.com/locate/jmaa Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter Paul A. Binding, a,1 Patrick J. Browne, b,1 and Bruce A. Watson c,∗,2 a Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4 b Mathematical Sciences Group, Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6 c School of Mathematics, University of the Witwatersrand, Private Bag 3, PO WITS 2050, South Africa Received 13 June 2002 Submitted by J.R. McLaughlin Abstract Three inverse problems for a Sturm–Liouville boundary value problem −y ′′ + qy = λy, y(0) cos α = y ′ (0) sin α and y ′ (1) = f (λ)y(1) are considered for rational f . It is shown that the Weyl m-function uniquely determines α, f , and q , and is in turn uniquely determined by either two spectra from different values of α or by the Prüfer angle. For this it is necessary to produce direct results, of independent interest, on asymptotics and oscillation.  2003 Elsevier Inc. All rights reserved. Keywords: Sturm–Liouville; Eigenparameter dependent boundary conditions 1. Introduction Broadly, inverse spectral theory for Sturm–Liouville problems seeks information about the original problems in terms of spectral constructions that they generate. Particular con- structions of interest here will be Weyl’s m-function, Prüfer’s angle, and sequences of * Corresponding author. E-mail address: batson@maths.wits.ac.za (B.A. Watson). 1 Research supported in part by grants from the NSERC of Canada. 2 Research conducted while visiting University of Calgary and University of Saskatchewan and supported in part by the Centre for Applicable Analysis and Number Theory. 0022-247X/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.11.025