Canonical forms for boundary conditions of self-adjoint differential operators

Canonical forms of boundary conditions are important in the study of the eigenvalues of boundary conditions and their numerical computations. The known canonical forms for self-adjoint differential operators, with eigenvalue parameter dependent boundary conditions, are limited to 4-th order differential operators. We derive canonical forms for self-adjoint $2n$-th order differential operators with eigenvalue parameter dependent boundary conditions. We compare the 4-th order canonical forms to the canonical forms derived in this article.     

辛矩阵和四阶微分算子自共轭边界条件的基本型

给出了辛矩阵的定义,讨论了它的性质,并通过使用辛矩阵的方法研究四阶自共轭的边界条件,得到了四阶自共轭边界条件的基本型,从而使得其它各种自共轭的边界条件都可以通过基本型的辛变换得到.     

New canonical forms of self-adjoint boundary conditions for regular differential operators of order four

In this paper, we find new canonical forms of self-adjoint boundary conditions for regular differential operators of order two and four. In the second order case the new canonical form unifies the coupled and separated canonical forms which were known before. Our fourth order forms are similar to the new second order ones and also unify the coupled and separated forms. Canonical forms of self-adjoint boundary conditions are instrumental in the study of the dependence of eigenvalues on the boundary conditions and for their numerical computation. In the second order case this dependence is now well understood due to some surprisingly recent results given the long history and voluminous literature of Sturm-Liouville problems. And there is a robust code for their computation: SLEIGN2.     

四阶正则微分算子耦合自共轭边界条件的基本标准型

本文利用新的方法给出了4阶正则微分算子耦合自共轭边界条件的基本标准型,新标准型中的4个分块小矩阵为对称矩阵,且其行列式的模为1.这与2阶微分算子耦合边界条件的标准型极为类似,这为给出一般的高阶微分算子自共轭边界条件标准型提供了新的思路.     

The classification of self-adjoint boundary conditions of differential operators with two singular endpoints

For general even order linear ordinary differential equations with real coefficients and endpoints which are regular or singular and for arbitrary deficiency index d, the self-adjoint domains are determined by d linearly independent boundary conditions. These conditions are of three types: separated, coupled, and mixed. We give a construction for all conditions of each type and determine the number of conditions of each type possible for a given self-adjoint domain. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm-Liouville, case. Even for regular problems of order four no such forms are available. In the case when all d conditions are separated this construction yields explicit non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case.     

The classification of self-adjoint boundary conditions: Separated, coupled, and mixed

There are three basic types of self-adjoint regular and singular boundary conditions: separated, coupled, and mixed. For even order problems with real coefficients, one regular endpoint and arbitrary deficiency index d, we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square-integrable solutions for some real value of the spectral parameter. In the separated case our construction yields non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm-Liouville, case. Even for regular problems of order four no such forms are available.     

On the singular nonlinear self-adjoint eigenvalue problem for differential algebraic systems of equations

The general nonlinear self-adjoint eigenvalue problem for a differential algebraic system of equations on a half-line is examined. The boundary conditions are chosen so that the solution to this system is bounded at infinity. Under certain assumptions, the original problem can be reduced to a self-adjoint system of differential equations. After certain transformations, this system, combined with the boundary conditions, forms a nonlinear self-adjoint eigenvalue problem. Requirements for the appropriate boundary conditions are clarified. Under the additional assumption that the initial data are monotone functions of the spectral parameter, a method is proposed for calculating the number of eigenvalues of the original problem that lie on a prescribed interval of this parameter.     

常型Sturm-Liouville问题的左定边值条件

本文刻画了常型Sturm-Liouville问题的左定边值条件．通过Sturm-Liouville微分算式的系数、区间端点以及边值条件给出了其左定性的充要条件．应用自伴边值条件分类,确切地给出了所有可能的左定边值条件．     

奇型Sturm-Liouville微分算子的限界自伴扩张

本文给出了奇型Sturm—Liouville微分算子限界自伴扩张的充要条件,从而得 到按边值条件分类的所有限界自伴边值条件,并直接回答了奇型Sturm—Liouville问题 的最小特征值不等式中相等的边值条件.     

常型Sturm-Liouville问题的左定空间

本文刻画了常型Sturm-Liouville问题的左定空间的一般形式．根据自伴边值条件的分类,确切地给出了所有可能的左定空间描述．     

Regularity of self-adjoint boundary conditions

It is shown that self-adjoint boundary conditions for ordinary differential operators of odd order are regular in Birkhoff's sense. A similar result, for differential operators of even order, was proved by a different method by Salaff. In short, Kamke's hypothesis about the regularity of self-adjoint boundary conditions is completely confirmed.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 835–846, December, 1977.In conclusion, the author thanks A. P. Khromov for suggesting the problem and guidance.     

Zur Regularität Selbstadjungierter Randwertaufgaben

In this note we derive a necessary requirement for the self-adjointness of a boundary value problem of ordinary differential equations, which involves only the leading coefficients of the normalized boundary conditions. As application it is shown that every self-adjoint boundary value problem of even order (and thus especially every real self-adjoint boundary value problem) is regular. The proof of this fact depends on the evaluation of a sort of generalized Vandermonde's determinant.     

Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.     

Quantum Graphs with Small Edges: Holomorphy of Resolvents

Doklady Mathematics - We consider a general scalar self-adjoint elliptic second order operator with general boundary conditions on an arbitrary metric graph containing a subgraph with edges of...     

一般边界条件下Sturm-Liouville算子的Ambarzumyan型定理

该文研究有限区间上一般自伴边界条件下的Sturm-Liouville方程的逆特征值问题.将Neumann边界条件下Sturm-Liouville方程的Ambarzumyan型定理推广到一般自伴边界条件下情形,证明了如果它的特征值与零势的特征值一样,则Sturm-Liouville方程的势为零.     

左定Sturm-Liouville算子的特征值计算(英文)

研究了定义在有限区间[a,b]上的具有分离型和混合型边界条件的左定正则Sturm-Liouville算子的特征值问题.把具有混合型边界条件的左定正则Sturm-Liouville问题转化成二维的、具有分离型边界条件的右定正则Sturm-Liouville问题,给出了具有混合型边界条件的左定正则Sturm-Liouville算子的特征值的数值计算方法.     

Eigenvalues of second-order difference equations with coupled boundary conditions

This paper is concerned with coupled boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues is proved, numbers of eigenvalues are calculated, and relationships between the eigenvalues of a self-adjoint second-order difference equation with three different coupled boundary conditions are established. These results extend the relevant existing results of periodic and antiperiodic boundary value problems.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

Knotted linear force-free magnetic fields-topological aspects

The problem of computing linear force-free magnetic fields on a knotted multiply-connected domain is considered. The domain is the support of the current distribution, and the linear force-free fieldproblem reduces to finding an eigenfield of a self-adjoint curl operator. In this context, the GKN Theorem is reformulated in terms of symplectic geometry in order to characterize the self-adjoint extensions of the curl operator restricted to solenoidal vector fields. When further restricted to the isotopy invariant boundary conditions, the self-adjoint extensions are parametrized by the Lagrangian subspaces of the symplectic form on the first homology group of the boundary. This paper discusses some of the topological aspects and gives some pointers for the associated finite element discretization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)