Inverse Spectral Problems for Sturm–Liouville Equations with Eigenparameter Dependent Boundary Conditions

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 l y = (1/ r )(−( py ′)′+ qy ),l ˜ y = ( 1 / r ˜) ( ‐( p ˜ y ′ ) ′ + q ˜ y ) ,Δ = [a bc d]and∑ = [r st 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 = λ y with boundary conditions y (0)cosα+ y ′(0)sinα = 0 and ( a λ+ b ) y (1) = ( c λ+ 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 λ n 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 ∫| 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 ˜ .