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.     

Similarity of Some Singular Operators to Self-Adjoint Ones

The singular differential operator is studied. It is proved that if the second moment of p is finite and L has no nonreal eigenvalues, then L is similar to a self-adjoint operator. The proof is based on an integral resolvent criterion of similarity applied to a wide class of functions p(x). Bibliography: 20 titles.     

Similarity of $$sgn x( - \tfrac{{d^2 }}{{dx^2 }} + c\delta )$$ Type Operators to Normal and Self-adjoint Operators

Mathematical Notes -     

对称常微分算子的两种自伴延拓形式之间的联系

建立了对称常微分算子von N eum ann自伴延拓与以边条件形式表达的自伴延拓之间的对应关系,给出了相互转换的方法.     

Spectral Analysis of Certain Self-Adjoint Differential Operators in a Space with Indefinite Metric

A second-order differential operator self-adjoint with respect to an indefinite metric on the circle is considered. The spectral resolution of this operator is found. The hypergeometric function is used in the computation of the Plancherel measure.     

奇型微分算子极大增生性的解析刻画

考虑奇型实系数微分算式ｌ（ｙ）＝（－１）~ｊ（ｐｎ（ｘ）ｙ（ｊ）（ｘ）在函数空间Ｌ２［ａ］上,本文借助边界条件刻画了ｌ（ｙ）生成极大增生算子的充要条件,及其ｌ（ｙ）最小生成算子为非负自伴扩张的充要条件．     

微分方程解刻画的实系数对称微分算子的自共轭域

通过把两个奇异端点的边界条件加以分离,利用微分方程的解(实参数解或复参数解)给出了实系数对称微分算子最大算子域的一种新的分解.进而应用这些解统一对其自共轭域进行描述,给出了自共轭域的完全刻画.     

两个四阶微分算子积的自伴性

本文研究由微分算式D^4-DpD q生成的两个四阶微分算子Li(i=1，2)的积L2L1的自伴性，并在常型和奇型情形下，分别获得了两个四阶微分算子积自伴的充要条件．同时证明了若L1和L2自伴，则L=L2L1自伴的充要条件是L1=L2．     

亏指数为可数无穷的对称微分算子的自伴扩张

应用Ｈｉｌｂｅｒｔ空间内对称算子自伴域一种新的描述方法,得到直和空间内亏指数为可数无穷的对称微分算子自伴扩张的完全解析描述。     

具有负指数系数的微分算子的谱

本文研究了形如∑_n~k=o~((α_k)(e~((α_k)x))D~k(a_k≤0)及∑_k~n=o((-1)~k)α_(2k)D~ke~(α_(2k)x)D~k+i/2∑_k~n=o(α_(2k+1))(D~ke~((α_(2k+1))x)D~(k+1)+D~(k+1)e~((α_(2k+1)x)D~k)(α_k≤0)的算式的谱问题,分别得到了它们的本质谱或本质谱所在的范围.     

自伴算子空间上保持Jordan积投影的线性映射

Let Bs(H) be the real linear space of all self-adjoint operators on a complex Hilbert space H with dim H ≥ 2.It is proved that a linear surjective map on Bs (H) preserves the nonzero projections of Jordan products of two operators if and only if there is a unitary or an anti-unitary operator U on H such that (X)=λU XU,X∈Bs(H) for some constant λ with λ∈{1,1}.     

具有内部奇异点的实系数微分算子的自共轭域

研究了一类带有内部奇异点的实系数微分算子自共轭域的描述问题.通过构造相应的直和空间,应用直和空间的相关理论及对相应最大算子域进行分解,在直和空间上生成的相应最小算子具有实正则型域的情形下,利用微分方程的实参数解给出此类算子的自共轭域的完全解析描述,并且确定其边界条件的矩阵仅由微分方程的解在正则点的初始值决定.     

一类具指数系数的对称微分算子谱的离散性

研究了具指数函数系数的2n阶实系数微分算式生成的对称微分算子,利用算子的直和分解法及不等式估计得到此类微分算子谱是离散的充分条件.     

On Higher-Order Differential Operators with Singularity inside the Interval

Mathematical Notes -     

Several Formulas for the First Regularized Trace of Discrete Operators

We derive several abstract formulas (of Gelfand--Levitan type) for the first regularized trace of discrete operators under various conditions convenient for verification. The proof of the main result is based on the contour integration method with some modifications from analytic perturbation theory and from the approach proposed earlier by M. Dostani. Some known trace formulas are generalized and improved. We give examples of the applications of our abstract trace formulas for obtaining special trace formulas for ordinary differential operators and for partial differential operators.     

Oscillation and Nonoscillation of Higher Order Self-Adjoint Differential Equations

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation (-1)ⁿ(t^alphay⁽ⁿ⁾)⁽ⁿ⁾+q(t)y = 0 (*) are established. In these criteria, equation (*) is viewed as a perturbation of the conditionally oscillatory equation (-1)ⁿ(t^alphay⁽ⁿ⁾)⁽ⁿ⁾ - µ,t^2n-y = 0, where _n, is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.