Pencils of differential operators containing the eigenvalue parameter in the boundary conditions
Marco Marletta, Андрей Андреевич Шкаликов, Christiane Tretter · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 2003
The paper deals with linear pencils N − λ P of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs ( W , T ) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L 2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D ( T ) and W .