INVERSE SPECTRAL PROBLEMS FOR STURM–LIOUVILLE EQUATIONS WITH EIGENPARAMETER DEPENDENT BOUNDARY CONDITIONS

Abstract

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), l˜y = (1/r˜)(−(p˜y′)′+q˜y),

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where det Δ = δ > 0, c ≠ 0, det [sum ] > 0, t ≠ 0 and (cs + dr − au − tb)² < 4(cr − ta)(ds − ub). Denote by (l; α; Δ) the eigenvalue problem ly = λy with boundary conditions y(0)cosα+y′(0)sinα = 0 and (aλ+b)y(1) = (cλ+d)(py′)(1). Define (l˜; α; Δ) as above but with l replaced by l˜. Let w_n denote the eigenfunction of (l; α; Δ) having eigenvalue λn and initial conditions w_n(0) = sin α and pw′_n(0) = −cos α and let γ_n = −aw_n(1)+cpw′_n(1). Define w˜_n and γ˜_n similarly.

As sample results, it is proved that if (l; α; Δ) and (l˜; α; Δ) have the same spectrum, and (l; α; Σ) and (l˜; α; Σ) have the same spectrum or ∫¹₀[mid ]w_n[mid ]²rdt+([mid ]γ_n[mid ]²/δ) = ∫¹₀[mid ]w˜_n[mid ]²r˜dt+([mid ]γ˜_n[mid ]²/δ) for all n, then q/r = q˜/r˜.