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Inverse Spectral Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions |
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| Authors: | Binding P A; Browne P J; Watson B A |
| Institution: | 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 |
| Abstract: | Inverse Sturm–Liouville problems with eigenparameter-dependentboundary conditions are considered. Theorems analogous to thoseof both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(–(py')'+qy),  ,
 where det  =  > 0, c  0, det  > 0, t 0 and (cs + dr –au – tb)2 < 4(cr – ta)(ds – ub). Denoteby (l;  ; ) the eigenvalue problem ly =  y with boundary conditionsy(0)cos+y'(0)sin = 0 and (a+b)y(1) = (c+d)(py')(1). Define ( ; ; ) as above but with l replacedby . Let wn denote the eigenfunctionof (l; ; ) having eigenvalue n and initial conditions wn(0)= sin and pw'n(0) = –cos and let  n = –awn(1)+cpw'n(1).Define  n and  n similarly. As sample results, it is proved that if (l; ; ) and (; ; ) have the same spectrum, and (l;;  ) and (; ; ) have the samespectrum or  for all n, thenq/r =  / . |
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