Sturm–Liouville Problems with Transfer Condition Herglotz Dependent on the Eigenparameter: Eigenvalue Asymptotics

• Published in: Complex analysis and operator theory 2021-06, Vol.15 (4), Article 71
• Main Authors: Bartels, Casey, Currie, Sonja, Watson, Bruce A.
• Format: Article
• Language: English
• Subjects: Analysis; Asymptotic properties; Boundary conditions; Boundary value problems; Eigenvalues; Hilbert space; Liouville equations; Mathematics; Mathematics and Statistics; Operator Theory
• ISSN: 1661-8254; 1661-8262
• DOI: 10.1007/s11785-021-01119-1

We consider a Sturm–Liouville equation ℓ y : = - y ′ ′ + q y = λ y on the intervals ( - a , 0 ) and (0,  b ) with a , b > 0 and q ∈ L 2 ( - a , b ) . Boundary conditions y ( - a ) cos α = y ′ ( - a ) sin α , y ( b ) cos β = y ′ ( b ) sin β , where α ∈ [ 0 , π ) and β ∈ ( 0 , π ] , are imposed, together with transmission conditions rationally-dependent on the eigenparameter via - y ( 0 + ) λ η - ξ - ∑ i = 1 N b i 2 λ - c i = y ′ ( 0 + ) - y ′ ( 0 - ) , y ′ ( 0 - ) λ κ + ζ - ∑ j = 1 M a j 2 λ - d j = y ( 0 + ) - y ( 0 - ) , with b i , a j > 0 for i = 1 , … , N , and j = 1 , ⋯ , M . Here we take η , κ ≥ 0 and N , M ∈ N 0 . In Bartels et al. (Integr Equ Oper Theory 90(3):1–20, 2018), it was shown that this boundary value problem can be posed as a self-adjoint operator eigenvalue problem in a suitable Hilbert space. In the present work we develop eigenvalue asymptotics for this class of problem.