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.     

A simple bound for the remainder of the neumann series in the case of a self-adjoint compact operator

We consider the functional equation of the second kind \gf − λK\gf = f with K a compact self-adjoint linear operator on a Hilbert space: a Fredholm integral equation of the second kind, for example. We establish the simple bound where λ is any regular value of K; ø is the solution of the equation corresponding to λ; λ₁ is the characteristic value of K smallest in absolute value; and N = 0, 1, 2, .... For \s|λ\s| < \s|λ₁\s|, this is an estimate for the remainder of the partial sums of the Neumann series.     

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n_±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A^?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.     

一类没有单调性假设的渐近线性波方程的多解

本文借助文[1]除去零空间的技巧,利用L—S度数理论讨论了一类浙近线性波方程。在对非线性项不做单调性假设的前提下,给出了一个新的多解定理。     

Inverse problems with partial data for a Dirac system: A Carleman estimate approach

We prove that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary. The proof is based on a combination of Carleman estimates for first and second order systems, and involves a reduction of the boundary measurements to the second order case. For this reduction a certain amount of decoupling is required. To effectively make use of the decoupling, the Carleman estimates are established for coefficients which may become singular in the asymptotic limit.     

2n阶微分算子乘积自伴的充分必要条件

给定Hilbert空间L2[a,∞)上两个由2n阶对称微分算式生成的微分算子Li(i=1,2),该文给出了乘积算子L2L1是自伴算子的一个充分必要条件．     

一类自伴微分算子谱的离散性

本文研究了２ｎ阶实系数Ｅｕｌｅｒ微分算式生成的对称微分算子,得到了自伴Ｅｕｌｅｒ微分算子的谱是离散的充分必要条件．     

Heisenberg operators of a Dirac particle interacting with the quantum radiation field

We consider a quantum system of a Dirac particle interacting with the quantum radiation field, where the Dirac particle is in a 4×4-Hermitian matrix-valued potential V. Under the assumption that the total Hamiltonian HV is essentially self-adjoint (we denote its closure by ), we investigate properties of the Heisenberg operator (j=1,2,3) of the j-th position operator of the Dirac particle at time t∈R and its strong derivative dxj(t)/dt (the j-th velocity operator), where xj is the multiplication operator by the j-th coordinate variable xj (the j-th position operator at time t=0). We prove that D(xj), the domain of the position operator xj, is invariant under the action of the unitary operator for all t∈R and establish a mathematically rigorous formula for xj(t). Moreover, we derive asymptotic expansions of Heisenberg operators in the coupling constant q∈R (the electric charge of the Dirac particle).     

Lower bound for the spectrum and the presence of pure point spectrum of a singular discrete Hamiltonian system

In this paper, criteria for limit-point (n) case of a singular discrete Hamiltonian system are established. Furthermore, the lower bound of the essential spectrum is obtained and the present of pure point spectrum is discussed for such system by using the spectral theory of self-adjoint operators in a Hilbert space.     

Spectral asymptotics and regularized traces for Dirac operators on a star-shaped graph

In this article, we consider the spectral problems for Dirac operators on a star-shaped graph. First, we derive the asymptotic expressions of the eigenvalues and then explicitly establish the regularized trace formulae for the operators in terms of the coefficients of the operators.     

Semi-Classical Green Kernel Asymptotics for the Dirac Operator

We consider a semi-classical Dirac operator in d ∈ ? spatial dimensions with a smooth potential whose partial derivatives of any order are bounded by suitable constants. We prove that the distribution kernel of the inverse operator evaluated at two distinct points fulfilling a certain hypothesis can be represented as the product of an exponentially decaying factor involving an associated Agmon distance and some amplitude admitting a complete asymptotic expansion in powers of the semi-classical parameter. Moreover, we find an explicit formula for the leading term in that expansion.     

高阶常型微分算子自伴域的辛几何刻划

考虑高阶常型实系数微分算子ι(y)=n↑∑↑k=0(pn-ky^(κ))^(κ)(x∈[α,b])。利用辛几何，对ι（y）的自伴域进行了分类，给出了ι（y）自伴域是κ－级的充要条件（0≤κ≤n）。     

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.     

A Sobolev-like inequality for the Dirac operator

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.     

A local lifting theorem for subnormal operators

Criteria for the existence of lifts of operators intertwining subnormal operators are established. The main result of the paper reduces lifting questions for general subnormal operators to questions about lifts of cyclic subnormal operators. It is shown that in general the existence of local lifts (i.e. those coming from cyclic parts) for a pair of subnormal operators does not imply the existence of a global lift. However this is the case when minimal normal extensions of subnormal operators in question are star-cyclic.

     

On a conformal invariant of the dirac operator on noncompact manifolds

We extend a Yamabe-type invariant of the Dirac operator to noncompact manifolds and show that as in the compact case this invariant is bounded by the corresponding invariant of the standard sphere. Further, this invariant will lead to an obstruction of the conformal compactification of complete noncompact manifolds. Mathematics subject classifications (2000): Primary 53C27, Secondary 53C21     

关于α次的殆星映射推广的Roper-Suffridge算子的一个注记

Suppose f is an almost starlike function of order α on the unit disk D. In this paper, we will prove that Φn, β2, γ2, …, βn, γn (f)(z) = (f(z1), (f(z1)/z1)β2(f'(z1))γ2z2,…,(f(z1)/z1)βn(f'(z11))γnzn)' preserves almost starlikeness of order α on Ωn,p1,p2,…,pn = {z =(z1,z2,…,zn)' ∈ Cn n∑j=1 |zj|pj ＜ 1}, where 0 ＜ p11 ≤ 2, pj ≥ 1, j = 2,…,n, are real numbers.