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Inverse Spectral Problems for Sturm–Liouville Equations with Eigenparameter Dependent Boundary Conditions
Author(s) -
Binding P. A.,
Browne P. J.,
Watson B. A.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700008899
Subject(s) - eigenfunction , eigenvalues and eigenvectors , inverse , spectrum (functional analysis) , sturm–liouville theory , boundary (topology) , combinatorics , boundary value problem , mathematics , spectral properties , physics , mathematical analysis , geometry , quantum mechanics , astrophysics
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 ˜ .