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We consider Sturm-Liouville boundary-value problems on the interval [0, 1] of the form −y + qy = λy with boundary conditions y (0) sin α = y(0) cos α and y (1) = (aλ + b)y(1), where a < 0. We show that via multiple Crum-Darboux transformations, this boundary-value problem can be transformed 'almost' isospectrally to a boundary-value problem of the same form, but with the boundary condition at x = 1 replaced by y (1) sin β = y(1) cos β, for some β.

We study Sturm-Liouville problems with right-hand boundary conditions depending on the spectral parameter in a quadratic manner. A modified Crum-Darboux transformation is used to produce chains of problems almost isospectral with the given one. The problems in the chain have boundary conditions which in various cases are affine or bilinear in the spectral parameter, and in all cases culminate in a problem with constant boundary conditions. This extends recent work of Binding, Browne, Code and Watson when the right-hand condition is either an affine function of the spectral parameter with negative leading coefficient or a Herglotz function.

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