Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients

Aleksandr Kholkin

doi:10.5772/intechopen.95820

Abstract

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green’s function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.

Highlights

The question of the generalization of the oscillatory Sturm theorem for scalar equations of higher orders and for equations with matrix coefficients for a long time remained open
A Sturm-type oscillation theorem was proved [2] for a problem on finite and infinite intervals for a second-order equation with block-triangular matrix coefficients
In the study of the connection between spectral and oscillation properties of non-self-adjoint differential operators with block-triangular operator coefficients [2, 4] the question arises of the structure of the spectrum of such operators

Summary

Introduction

The question of the generalization of the oscillatory Sturm theorem for scalar equations of higher orders and for equations with matrix coefficients for a long time remained open. In the study of the connection between spectral and oscillation properties of non-self-adjoint differential operators with block-triangular operator coefficients [2, 4] the question arises of the structure of the spectrum of such operators. 2. The fundamental solutions for an non-self-adjoint differential operator with block – triangular operator coefficients. We suppose that coefficients of the Eq (1) satisfy relations: Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular. 2.1 Construction of the fundamental system of solutions for an operator differential equation with a rapidly increasing at infinity potential. X!∞ γ∞ðx, λÞ : consider another block-triangular operator solution Ψ~ ðx, λÞ that increases at infinity diagonal blocks which are defined by. □. the fundamental system of solution is constructed for an operator differential equation with a rapidly increasing at infinity potential

Asymptotic of the fundamental system solutions of equation with block-triangular potential

Conclusion