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 (−π,−π+d),(−π+d,π−d) and (π−d,π) together with eigenparameter dependent boundary conditions and with additional transmission conditions specified at the common end points −π+d and π−d, where 0<d<π. 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.

Keywords

• boundary-value-transmission problems
• eigenvalues
• generalized eigenfunctions
• lower bound estimation
• Rayleigh’s method
• transmission conditions