Basis Property of Eigenfunctions in Lebesgue Spaces for a Spectral Problem with a Point of Discontinuity

Resumo

We study the basis properties of eigenfunctions in Lebesgue spaces for a spectral problem for a discontinuous second-order differential operator with spectral parameter in the discontinuity (transmission) conditions. This problem arises when solving the problem on the vibrations of a loaded spring with fixed endpoints. An abstract theorem on the stability of basis properties of multiple systems in a Banach space with respect to certain transformations is proved. This theorem is used when proving theorems on the basis property of eigenfunctions of a discontinuous differential operator in the Lebesgue spaces L_p ⊕ ℂ and L_p.