Journal of Lie Theory 23 (2013), No. 4, 921--935
Copyright Heldermann Verlag 2013
On the Isospectral Sixth Order Sturm-Liouville Equation
Kazem Ghanbari
Mathematics Department, Sahand University of Technology, Tabriz, Iran
kghanbari@sut.ac.ir
Hanif Mirzaei
Mathematics Department, Sahand University of Technology, Tabriz, Iran
h.mirzaei@sut.ac.ir
We investigate families of sixth-order Sturm-Liouville equations having the same spectrum. We factorize the Sturm-Liouville operator as the product of a third order linear differential operator and its adjoint. By reversing the order of the factors we obtain another sixth-order Sturm-Liouville operator which is isospectral with the initial operator. The factorization is possible provided the coefficients of the factors satisfy a system of nonlinear third-order ordinary differential equations so called principal system. The coefficients in the factorization products are solutions of the principal system. We study this system by using Lie group of symmetries and we show that it may admit a one or two parameter Lie group of transformations. One of the cases leads to Chazy's equation which admits a three parameter Lie group of transformations. In some cases, we solve the system and obtain an isospectral operator.
Keywords: Sixth order Sturm-Liouville equation, isospectral, Lie group symmetries.
MSC: 34B24, 70G65
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