On the completeness property of root vector systems for 2 × 2 Dirac type operators with non-regular boundary conditions

IF 1.2 3区 数学 Q1 MATHEMATICS
Anton A. Lunyov , Mark M. Malamud
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引用次数: 0

Abstract

The paper is concerned with the completeness property of a system of root vectors of a boundary value problem for the following 2×2 Dirac type equationLy=iB1y+Q(x)y=λy,y=col(y1,y2),x[0,1],B=diag(b1,b2),b1<0<b2,andQW1n[0,1]C2×2, subject to general non-regular two-point boundary conditions Cy(0)+Dy(1)=0. If b2=b1=1, this equation is equivalent to the one dimensional Dirac equation.
We establish a new completeness result for the system of root vectors of such boundary value problem with non-regular and even degenerate boundary conditions. We also present several explicit completeness results in terms of values Q(j)(0) and Q(j)(1). In the case of degenerate boundary conditions and the analytic Q(), the criterion of the completeness property is established. We demonstrate our results on the explicit example of a complete system of vector quasi-exponential polynomials.
Applications to the spectral synthesis for Dirac type operators are discussed. Moreover, applications to the completeness property for the damped string equation are provided.
论具有非规则边界条件的 2 × 2 迪拉克型算子根向量系统的完备性
本文关注以下 2×2 狄拉克方程的边界值问题根向量系统的完备性Ly=-iB-1y′+Q(x)y=λy,y=col(y1,y2),x∈[0,1],B=diag(b1,b2),b1<;0<b2,Q∈W1n[0,1]⊗C2×2,受一般非规则两点边界条件 Cy(0)+Dy(1)=0。如果 b2=-b1=1, 这个方程等价于一维狄拉克方程。我们为这种具有非规则甚至退化边界条件的边界值问题的根向量系统建立了一个新的完备性结果。我们还以值 Q(j)(0)和 Q(j)(1)为单位提出了几个显式完备性结果。在退化边界条件和解析 Q(⋅) 的情况下,我们建立了完备性准则。我们以矢量准指数多项式的完整系统为例演示了我们的结果。此外,我们还提供了阻尼弦方程完备性的应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.50
自引率
7.70%
发文量
790
审稿时长
6 months
期刊介绍: The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.
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