Spectral properties of the Helmholtz problem with spectral parameter Dependent conditions

Abstract

We consider Helmholtz problems containing a spectral parameter both in the equation and in the boundary conditions. we prove that the system of corresponding eigen functions forms an orthonormal basis in some adequate Hilbert spaces. The oscillation properties as completeness, minimality and basic properties are investigated for the eigenfunction of the Helmholtz operator equation in the triple of adequate Hilbert spaces. Asymptotic formula for eigenvalue and eigenfunction are deduced.