Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied.     

对称算子自共轭扩张的酉参数化描述(英文)

利用构造性的方法,给出了边值空间理论中几个结果新的证明,其中,边值空间理论是有关对称算子自共轭扩张的一种方法.同时,得到了几个新的结果.如发现了一般的边界三元组所具有的结构.进一步地,利用这个结果证明了辅助Hilbert空间H上的酉变换与亏空间K-和K+之间的等距同构映射间存在一个双解析的映射.发现并证明了一般边界条件:B(ψ):=MΓ1ψ+NΓ2ψ=0(其中M,N是阶数为亏指数的方阵)是自共轭的充要条件以及相应的酉变换和边界映射.     

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

The Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems

In this paper, the Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems is developed. A minimal and a maximal operators, GKN-sets, and a boundary space for the system are introduced. Algebraic characterizations of the domains of self-adjoint extensions of the minimal operator are given. A close relationship between the domains of self-adjoint extensions and the GKN-sets is established. It is shown that there exist one-to-one correspondences among the set of all the self-adjoint extensions, the set of all the d-dimensional Lagrangian subspaces of the boundary space, and the set of all the complete Lagrangian subspaces of the boundary space.     

Singular point perturbation of an odd operator in ℤ2-Graded space

In the paper we study supersymmetric models for point interaction perturbations of operators of Dirac type and their spectral properties. Such models are considered in the class of odd self-adjoint operators in ℤ₂-graded Pontryagin space. We present in detail the previously considered realization method of strongly singular perturbation by means of their embedding into the theory of self-adjoint extensions. We describe odd self-adjoint extensions of odd symmetric operators with deficiency indices (1,1) in ℤ₂-graded Pontryagin space and squares of such extensions using Krein’s formula for the resolvent. The results obtained are refined in application to singular perturbations of odd self-adjoint differential operators. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 924–940, December, 1999.     

Triplet extensions I: Semibounded operators in the scale of Hilbert spaces

The extension theory for semibounded symmetric operators is generalized by including operators acting in a triplet of Hilbert spaces. We concentrate our attention on the case where the minimal operator is essentially self-adjoint in the basic Hilbert space and construct a family of its self-adjoint extensions inside the triplet. All such extensions can be described by certain boundary conditions, and a natural counterpart of Krein’s resolvent formula is obtained.     

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.     

On the negative squares of indefinite Sturm-Liouville operators

The number of negative squares of all self-adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a Krein space is characterized in terms of the behaviour of an abstract Titchmarsh-Weyl function near 0 and ∞. These results are applied to a large class of symmetric and self-adjoint indefinite Sturm-Liouville operators with indefinite weight functions.     

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.     

An unusual self-adjoint linear partial differential operator

In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region.

This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic' operator.

The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions.

In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty.

This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty.

Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent.

In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.

     

Some Remarks on Krein's Extension Theory

The extension problem of semibounded symmetric operators and symmetric operators with a gap is studied in detail. Using a suitable representation (Krein model) for the inverses of those operators a parameterization of their symmetric and self-adjoint extensions is introduced which improves Krein's famous extension theory. In particular, the parameterization clearly shows which self-adjoint extensions in the gap case correspond to Friedrichs and v. Neumann or Krein extensions in the semibounded case. Moreover, special properties of the extensions as the exactness of the gap are characterized in terms of the parameters.     

On semibounded restrictions of self-adjoint operators

After the von Neumann's remark [10] about pathologies of unbounded symmetric operators and an abstract theorem about stability domain [9], we develope here a general theory allowing to construct semibounded restrictions of selfadjoint operators in explicit form. We apply this theory to quantum-mechanical momentum (position) operator to describe corresponding stability domains. Generalization to the case of measurable functions of these operators is considered. In conclusion we discuss spectral properties of self-adjoint extensions of constructed self-adjoint restrictions.     

Nontrivial Attractors in a Model Related to the Three-Body Quantum Problem

A model of a quantum mechanical system related to the three-body problem is studied. The model is defined in terms of a symmetric pseudo-differential operator (PDO) with unbounded symbol. The entire family of self-adjoint extensions of this operator is studied using harmonic analysis. A regularization procedure for this PDO is introduced and the spectral properties of the operators obtained in this way are investigated. The limit behavior of the regularized operators when the regularization parameter is removed is analyzed and a nontrivial attractor is exhibited.     

Dissipative Dirac Operator with General Boundary Conditions on Time Scales

Ukrainian Mathematical Journal - We consider symmetric Dirac operators on bounded time scales. Under general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint,...     

Sturm-Liouville operators with measure-valued coefficients

We give a comprehensive treatment of Sturm-Liouville operators whose coefficients are measures, including a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh-Kodaira theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm-Liouville operators, Sturm-Liouville operators with (local and non-local) δ and δ′ interactions or transmission conditions as well as eigenparameter dependent boundary conditions, Krein string operators, Lax operators arising in the treatment of the Camassa-Holm equation, Jacobi operators, and Sturm-Liouville operators on time scales as special cases.