Sturm–Liouville problems with boundary conditions depending quadratically on the eigenparameter

Abstract

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.

Spectral parameter power series for Sturm–Liouville problems

“…Eigenvalues of (22) for ε = 0.01 n λ n SPPS δ 1 δ 2 1 1.0001 1.0×10 −14 3.8×10 −16 2 2.0008 1.5×10 −13 1.7×10 −14 3 3.00269 3.0×10 −12 7.3×10 −14 4 4.00638 4.9×10 −11 2.9×10 −12 5 5.01243 6.8×10 −10 1.3×10 −11 6 6.02143 8.8×10 −9 6.5×10 −9 7 7.03393 3.2×10 −7 6.7×10 −8 8 8.05048 6.2×10 −6 3.4×10 −7 9…”

Eigenvalue problems, spectral parameter power series, and modern applications

“…In this case together with equation and boundary condition , the eigenfunction must satisfy a second boundary condition of the form where ϕ is a complex‐valued function of the variable λ and β 1 , β 2 , , are complex numbers. For some special forms of the function ϕ such as ϕ ( λ ) = λ or ϕ ( λ ) = λ 2 + c 1 λ + c 2 , results were obtained , concerning the regularity of the problem , , and ; we will not dwell upon the details. In general, the presence of the spectral parameter in boundary conditions introduces additional considerable difficulties both in theoretical and numerical analysis of the problems.…”

“…/ D or '. / D 2 C c 1 C c 2 , results were obtained [35], [38] concerning the regularity of the problem (21), (22), and (26); we will not dwell upon the details. In general, the presence of the spectral parameter in boundary conditions introduces additional considerable difficulties both in theoretical and numerical analysis of the problems.…”

Transmutations and Spectral Parameter Power Series in Eigenvalue Problems

“…where α is an arbitrary complex number, φ is a complex-valued function of the variable λ and β 1 , β 2 , β ′ 1 , β ′ 2 are complex numbers. For some special forms of the function φ such as φ(λ) = λ or φ(λ) = λ 2 + c 1 λ + c 2 , results were obtained [19], [64] concerning the regularity of the problem; we will not dwell upon the details. Notice that the SPPS approach is applicable as well to a more general Sturm-Liouville equation (pu ′ ) ′ + qu = λru.…”