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Journal of the London Mathematical Society 2000 62(1):161-182; doi:10.1112/S0024610700008899
© 2000 by London Mathematical Society
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© The London Mathematical Society

Inverse Spectral Problems for Sturm–Liouville Equations with Eigenparameter Dependent Boundary Conditions

P. A. Binding, P. J. Browne and B. A. Watson

Department of Mathematics and Statistics, University of Calgary Calgary, Alberta, Canada T2N 1N4
Department of Mathematics and Statistics, University of Saskatchewan Saskatoon, Saskatchewan, Canada S7N 5E6
Department of Mathematics, University of the Witwatersrand Private Bag 3, PO WITS 2050, South Africa

Received 10 November 1998. Revision received 25 March 1999.

Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved.

In particular, let ly = (1/r)(–(py')'+qy), Formula,


Formula

where det {Delta} = {delta} > 0, c != 0, det {sum} > 0, t != 0 and (cs + dr – au – tb)2 < 4(cr – ta)(ds – ub). Denote by (l; {alpha}; {Delta}) the eigenvalue problem ly = {lambda}y with boundary conditions y(0)cos{alpha}+y'(0)sin{alpha} = 0 and (a{lambda}+b)y(1) = (c{lambda}+d)(py')(1). Define (Formula; {alpha}; {Delta}) as above but with l replaced by Formula. Let wn denote the eigenfunction of (l; {alpha}; {Delta}) having eigenvalue {lambda}n and initial conditions wn(0) = sin {alpha} and pw'n(0) = –cos {alpha} and let {gamma}n = –awn(1)+cpw'n(1). Define Formulan and Formulan similarly.

As sample results, it is proved that if (l; {alpha}; {Delta}) and (Formula; {alpha}; {Delta}) have the same spectrum, and (l; {alpha}; {Sigma}) and (Formula; {alpha}; {Sigma}) have the same spectrum or Formula for all n, then q/r = Formula/Formula.


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