Operator-pencil treatment of multi-interval Sturm-Liouville equation with boundary-transmission conditions

Abstract

This paper is devoted to a new type of boundary-value problems for Sturm-Liouville equations defined on three disjoint intervals (􀀀 ;􀀀 + d); (􀀀 + d; 􀀀 d) and ( 􀀀 d; ) together with eigenparameter dependent boundary conditions and with additional transmission conditions specified at the common end points 􀀀 + d and 􀀀 d; where 0 < d < . The considered problem cannot be treated by known techniques within the usual framework of classical Sturm-Liouville theory. To establish some important spectral characteristics we introduced the polynomial-operator formulation of the problem. Moreover, we develop a new modification of the Rayleigh method to obtain lower bound of eigenvalues.