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 ) , Δ = [ a b c d ] and ∑ = [ r s t u ] where det Δ = δ > 0, c ≠ 0, det ∑ > 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 wn denote the eigenfunction of (l; α; Δ) having eigenvalue λn and initial conditions wn(0) = sin α and pw′n(0) = −cos α and let γn = −awn(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 ∫ | w n | 0 1 r d t + ( | γ n | 2 / δ ) = ∫ | w ˜ n | 0 1 r ˜ d t + ( | γ ˜ n | 2 / δ ) for all n, then q/r = q ˜ / r ˜ .