Abstract
In this work a Sturm-Liouville operator with piecewise continuous coefficient and spectral parameter in the boundary conditions is considered. The eigenvalue problem is investigated; it is shown that the eigenfunctions form a complete system and an expansion formula with respect to the eigenfunctions is obtained. Uniqueness theorems for the solution of the inverse problem with a Weyl function and spectral data are proved.MSC: 34L10, 34L40, 34A55.
Highlights
We consider the boundary value problem–y + q(x)y = λ ρ(x)y, ≤ x ≤ π, ( )U(y) := y ( ) + α – λ α y( ) =,V (y) := λ β y (π ) + β y(π ) – β y (π ) – β y(π ) =, where q(x) ∈ L (, π) is a real valued function, λ is a complex parameter, αi, βj, i =, j =, are positive real numbers and⎧ ⎨, ρ(x) = ⎩γ,≤ x < a, a < x ≤ π, where < γ =
The eigenparameter appears in the differential equation of the Sturm-Liouville problem and in the boundary conditions, are given in [ – ]. Spectral analyses of these problems are examined as regards different aspects in [ – ]
We investigate a Sturm-Liouville operator with discontinuous coefficient and a spectral parameter in boundary conditions
Summary
In Section , an expansion formula with respect to the eigenfunctions is obtained and Section contains uniqueness theorems for the solution of the inverse problem with a Weyl function and spectral data. For any eigenvalue λn the solutions ( ), ( ) satisfy the relation ψ(x, λn) = knφ(x, λn) and the normalized numbers of the boundary value problem ( )-( ) are given below: π αn := φ (x, λn)ρ(x) dx + α φ ( , λn) Lemma The eigenvalues of the boundary value problem ( )-( ) are simple, i.e.
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